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[latex]x^{a+2}\cdot{x^{3a-9}}[/latex]. Good news, the rule is still true if you have more than one variable, or a combination of variables and numerals. Zero exponents rule Zero exponents examples Zero exponents rule The base b raised to the power of zero is equal to one: b0 = 1 Zero exponents examples Five raised to the power of zero is equal to one: 5 0 = 1 Minus five raised to the power of zero is equal to one: (-5) 0 = 1 Zero to raised the power of zero is equal to one: 0 0 = 1 See also Exponent rules Exponent rules chart How Prodigy can help you teach exponents The exponent rules explain how to solve various equations that as you might expect have exponents in them. In exponential form, you would write the product as [latex]2^{7}[/latex]. To evaluate 2), you would apply the exponent to the 3 and the negative sign: [latex]\begin{array}{c}{\left(-3\right)}^{2}\\=\left(-3\right)\cdot\left(-3\right)\\={ 9}\end{array}[/latex]. This term is in its most simplified form. A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction barfrom numerator to denominator or vice versa. If youre hoping to become a math teacher at a public school in Massachusetts, then now is a good time to start planning. This is covered by the fraction exponent rule (described in detail later). When you raise the base B to the exponent E, you are multiplying B by itself E times. [latex]x^{3}=64[/latex] when [latex]x=4[/latex], 4) [latex]3^x[/latex] if [latex]x = 4[/latex], Substitute x = 4 into the exponent. Simplify Expressions Using the Power Rule of Exponents (Basic). Definition: The Negative Exponent Rule. 100 =? The exponent on this term is 3, and the base is x, the 2 is not getting the exponent because there are no parentheses that tell us it is. It is the fourth power of 5 to the second power. First, simplify the denominator using the product rule of exponents. Simplify exponential expressions. What happens if you divide two numbers in exponential form with the same base? Use all the rules of exponents covered so far in this chapter to simplify the following. [latex]5^{4}[/latex]is read as 5 to the fourth power. It means [latex]5\cdot5\cdot5\cdot5[/latex], or 625. Calculus for Business and Social Sciences Corequisite Workbook (Dominguez, Martinez, and Saykali), { "5.01:_Definition_of_a" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.02:_The_Product_Rule_for_Exponents" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.03:_The_Quotient_Rule_of_Exponents" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.04:_Zero_Exponent_Rule" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.05:_The_Negative_Exponent_Rule" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.06:_Power_Rule_For_Exponents" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.07:_The_power_of_a_product_rule_for_exponents" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.08:_Power_of_a_quotient_rule_for_exponents" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "5.09:_Rational_Exponents" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_The_Real_Numbers_and_the_Number_Line" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Cartesian_Coordinate_System" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Interval_Notation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Exponents_and_Exponent_Rules" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Absolute_Value" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Straight_Lines" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Polynomial_Operations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Rational_Expressions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Inequalities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "Zero exponent rule", "license:ccbysa", "showtoc:no", "autonumheader:yes2", "licenseversion:40", "authorname:dominguezetal", "program:oeri" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FApplied_Mathematics%2FCalculus_for_Business_and_Social_Sciences_Corequisite_Workbook_(Dominguez_Martinez_and_Saykali)%2F05%253A_Exponents_and_Exponent_Rules%2F5.04%253A_Zero_Exponent_Rule, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), ASCCC Open Educational Resources Initiative, Victoria Dominguez, Cristian Martinez, & Sanaa Saykali, \(\dfrac{d^5 }{d^{2+3 }}= \dfrac{d^5 }{d^5} = d^{55 }= d^{0} = 1\), \(\dfrac{(ab^2 )^7 }{(ab^2)^2 \cdot (ab^2)^4 (ab^2)}\). 3) [latex]x^{3}[/latex] if [latex]x=4[/latex]. The expression cant be simplified any further. Substitute 4 for the variable x.notice the how adding parentheses can change the outcome when you are simplifying terms with exponents. You can present the same pattern for other numbers, too. In other words, a number raised to the power of zero is 1: But why should B0 be 1? This leads us to the definition of zero as an exponent 107, \(x ^ { 0 } = 1\: x \neq 0\) It is important to note that \(0^{0}\) is indeterminate. Created by Sal Khan. For any real numbers [latex]a[/latex] and [latex]b[/latex] and any integer [latex]n[/latex], the power of a product rule of exponents states that. Use the quotient rule to simplify each expression. The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas: The exponent says how many times to use the number in a multiplication. I'm the go-to guy for math answers. Simplify expressions with exponents equal to zero. Write each of the following quotients with a single base. About Transcript Any non-zero number to the zero power equals one. Caution! at first. In section 5.3, the exponent of the number in the numerator was always greater than the exponent of the number in the denominator. If we have a base with an exponent and the expression is raised to an exponent, we can multiply the exponents. Therefore, we can write the rule as a =1. You substitute the value of the variable into the expression and simplify. Simplify each of the following products as much as possible using the power of a product rule. Lets simplify [latex]\left(5^{2}\right)^{4}[/latex]. You substitute the value of the variable into the expression and simplify. . The first expression does not include parentheses so you would apply the exponent to the integer 3 first, then apply the negative sign. We worked with expressions for which[latex]a>b[/latex] so that the difference [latex]a-b[/latex] would never be zero or negative. In this article, well talk about exponents and how they work. For any real number \(a\), The Zero Exponent Rule is the following, \[x^5 = x \cdot x \cdot x \cdot x \cdot x \nonumber \], \[\dfrac{x^5 }{x^5} =\dfrac{ x \cdot x \cdot x \cdot x \cdot x }{x \cdot x \cdot x \cdot x \cdot x }= \dfrac{\cancel{x \cdot x \cdot x \cdot x \cdot x }}{\cancel{x \cdot x \cdot x \cdot x \cdot x }}= 1 \nonumber \], \[\dfrac{x ^5 }{x^5} = x^{55 }= x^0=1 \nonumber \]. To simplify a power of a power, you multiply the exponents, keeping the base the same. What happens if you multiply two numbers in exponential form with the same base? And we saw above that the answer is 58. We worked with expressions for which a >b a > b so that the difference ab a b would never be zero or negative. We do make the assumption that the expression ???3xy+a??? Any number divided by itself is equal to one. [latex]\left(5x\right)3=8,000[/latex] when [latex]x=4[/latex]. If a nonzero base is raised to a fraction exponent, then the result is a radical with index equal to the denominator of the fraction and a radicand of the base raised to the numerator of the fraction. For example, [latex]\left(2^{3}\right)^{5}=2^{15}[/latex]. This lesson will cover how to find the power of a number or nonzero variable raised to the zero power. Mastering these basic exponent rules along with basic rules of logarithms (also known as "log rules") will make your study of algebra very productive and enjoyable. The consent submitted will only be used for data processing originating from this website. \(\dfrac{(ab^2 )^7 }{(ab^2)^{2+4+1}}= \dfrac{(ab^2 )^7}{ (ab^2)^7} = (ab^2 )^{77 }= (ab^2 )^0 = 1\). If we have a product raised to an exponent, we can rewrite it as a product of two terms, each with the same exponent. They are integers that are exponents. The product rule for exponents says that we can add the exponents. If so, please share it with someone who can use the information. Manage Settings If the exponential expression is negative, such as 34, it means - (3 3 3 3) or 81. That means, exponent refers to how many times a number multiplied by itself. To see how this is done, let us begin with an example. If we write this out, we have 4 times 4, times 4, times 4 as the numerator and 4 times 4, times 4, times 4 as the denominator. It is the fourth power of 5 to the second power. Exponents are used in many areas, from finance to scientific research but what do they really mean? For example, the notation [latex]5^{4}[/latex]can be expanded and written as [latex]5\cdot5\cdot5\cdot5[/latex], or 625. When you have a base raised to a fraction exponent, you can convert to a radical expression like this: An exponent can be a decimal. Dont worry, just remember that any nonzero real number raised to the power of zero is one, so, Just remember that any nonzero real number raised to the power of zero is one. So,[latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[/latex]. So, what is an exponent? The law that x m x n . [latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}[/latex], You can rewrite the expression as: [latex] \displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}[/latex]. In the next sections, you will learn how to simplify expressions that contain exponents. The constant 5, can be factored out to see common bases clearly. The result of raising B to the power of E will be the value BE = B*B*B**B (with B appearing E times). \( \dfrac{1}{ \sqrt{5}} \cdot \dfrac{y^3 }{y^3 }= \dfrac{1}{ \sqrt{5}} \cdot y ^{33 }= \dfrac{1}{ \sqrt{5}} \cdot y^0 = \dfrac{1}{ \sqrt{5}} \cdot 1 = \dfrac{1}{ \sqrt{5}}\). For example: If we divide two exponential terms with the same base, then we can combine them into one term with the same base and an exponent that is the difference of the two original exponents (top minus bottom). What would happen if [latex]a=b[/latex]? [latex]b^{5}[/latex]is read as b to the fifth power. It means [latex]{b}\cdot{b}\cdot{b}\cdot{b}\cdot{b}[/latex]. Lets rewrite the original problem differently and look at the result. For now consider \(0^0\) to be undefined. To clarifywhether a negative sign is applied before or after the exponent, here is an example. A base raised to an exponent is a shortcut for repeated multiplication. Product, Quotient, and Power Rule for Exponents If a factor is repeated multiple times, then the product can be written in exponential form xn. Read more. 10^0 = \text {?} Another useful result occurs if we relax the condition that [latex]a>b[/latex] in the quotient rule even further. However a zero exponent creates a bit of a puzzle because you cannot multiply a number zero times. Simplify expressions with negative exponents. Of course, there are some rules for exponents that you should know (such as what . Of course, for a base of B = 1, we get a result of 1 for any exponent E. The reason is that when we multiply a number by 1, we always get the same number back (this is one of the rules of multiplication). [latex]\left(5^{2}\right)^{4}[/latex]is a power of a power. In the following video you will see more examples of using the product rule for exponents to simplify expressions. will always equal ???1??? Mathematicians debate the value of 00. Write answers with positive exponents. However, using the associative property of multiplication, begin by simplifying the first two. For example, 0.6 = 3/5, so we would write: Some decimal exponents cannot be converted to a fractional exponent. When you raise the base B to the exponent E, you are multiplying B by itself E times. Return to the quotient rule. Many arithmetic operations like addition, subtraction, multiplication, and division can be conveniently performed in quick steps using the laws of exponents. is not equal to zero. Exponential notation was developed to write repeated multiplication more efficiently. Some of our partners may process your data as a part of their legitimate business interest without asking for consent. if ???a??? We know that 24 = 16, 23 = 8, 22 = 4, and 21 = 2. Simplify Expressions Using Exponent Rules (Power of a Quotient). Identify the exponent and the base in the following terms, then simplify: The exponent in this term is 2 and the base is 7. will always equal ???1??? Do not simplify further. For example, 6 is multiplied by itself 4 times, i.e. An exponent may be referred to a number or a variable raised to another number or variable. Helpful steps to simplify expressions with exponents Identify common bases. Step-by-step math courses covering Pre-Algebra through Calculus 3. This lesson will cover how to find the power of a number or nonzero variable raised to the zero power. The general form of zero exponent rule is given by: a 0 = 1 and (a/b) 0 = 1. First, evaluate anything in Parentheses or grouping symbols. We go through why any variable or number raised to the zero is equal to 1.Transcript/notesA number raised to zero, such as a raised to the zero, is equal to 1. If we have a quotient raised to an exponent, we can rewrite it as a quotient of two terms, each with the same exponent. For instance, consider [latex]{\left(pq\right)}^{3}[/latex]. Caution! Notice we get the same result by adding the three exponents in one step. isnt equal to ???0???. For any non zero real number a and any integer n, the negative exponent rule is the following. In an equation: We should avoid B = 0 when x, y, or x + y is negative (to prevent zero denominators in any of the terms above). The rule for zero as an exponent: Any nonzero real number raised to the power of zero is one, this means anything that looks like ???a^0??? Its still true that any nonzero real number raised to the power of zero is one and we know that ?? This concept can be generalized in the following way: For any number x and any integers a and b,[latex]\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}[/latex]. If the base is negative, then the . The result of raising B to the power of E will be the value BE = B*B*B**B (with B appearing E times). Answer: When there is a number or variable that is raise to a power, the number (or variable) is referred to as the base whereas the superscript number refers to the exponent or power. We made the condition that m > n so that the difference m n would never be zero or negative. Let's simplify (52)4. Think about the expression[latex]\left(2+3\right)^{2}[/latex]. This is true for any nonzero real number, or any variable representing a nonzero real number. Of course, there are some rules for exponents that you should know (such as what happens when you use an exponent of zero, or when you multiply two exponential terms with the same base). Lets look at dividing terms containing exponential expressions. If needed combine common bases using the product rule of exponents. So, no matter how many times we multiply 1 by itself, we get 1 as a result. I create online courses to help you rock your math class. A base raised to an exponent is a shortcut for repeated multiplication. If a nonzero base is raised to an exponent of zero, then the result is 1. We have two exponentiated terms with the same base, so we can multiply them together. In an equation: We must avoid B = 0, because otherwise we would get a zero denominator when E is positive. To simplify, expand the multiplication: [latex]\left(-5\right)^{2}=-5\cdot{-5}=25[/latex]. The rule states that any nonzero base raised to a power of 0 is equal to 1. I hope you found this article helpful. It could be a whole number, a fraction, a negative number, or a decimal. Consider the following expression. For example, (pi) is a nonrepeating, nonterminating decimal (a transcendental number), so it cannot be written as a fraction. Its still true that any nonzero real number raised to the power of zero is one so. This is a special case that is covered in advanced courses. Lets look at rules that will allow you to do this. [latex]x^{a+2}\cdot{x^{3a-9}}=x^{(a+2)+(3a-9)}=x^{4a-7}[/latex]. An example of data being processed may be a unique identifier stored in a cookie. Another word for exponent is power. Use the power rule to simplify each expression. The exponent on this terms is 2 and the base is [latex]-5[/latex]. To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. In an equation: We should avoid B = 0 or C = 0 when x is negative to prevent zero denominators. We will learn what to do when a term with apoweris raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent. x9 x9 d5 d2 d3 5(xy)3 (xy)3 y3 5y3 (ab2)7 (ab2)2 (ab2)4(ab2) Solution Note: 00 does not equal 1. [latex]10^{3}[/latex] is read as 10 to the third power or 10 cubed. It means [latex]10\cdot10\cdot10[/latex], or 1,000. In the following video you will see examples of simplifying expressions with negative exponents. We can extend the idea of exponents to include zero and negative exponents as well. Write answers with positive exponents. Write each of the following products with a single base. The following examples show how to identify the base and the exponent, as well as how to identify the expanded and exponential format of writing repeated multiplication. Well, we can prove this to ourselves with an example. Simplify. In this case, the base is [latex]5^2[/latex]and the exponent is 4, so you multiply [latex]5^{2}[/latex]four times: [latex]\left(5^{2}\right)^{4}=5^{2}\cdot5^{2}\cdot5^{2}\cdot5^{2}=5^{8}[/latex](using the Product Ruleadd the exponents). In this case, you add the exponents. Just remember that any nonzero real number raised to the power of zero is one, so, Look different? Example 1 (-3) 0 = 1 (2/3) 0 = 1 0 = undefined. In that case, a decimal exponent is essentially the same as a fraction exponent. Lets use B = 2 for this case. Take a moment to contrast how this is different from the product rule for exponents found on the previous page. Do not try to apply this rule to sums. Simplify Expressions Using Exponent Rules (Power of a Product). 2) By following a pattern of decreasing an exponent by one by dividing by the base, we find that when we get to the 0 power, we end up dividing the base by itself, resulting in 1. Likewise, ( x)4 = ( x) ( x) ( x) ( x) = x4, while x4 = - (x x x x). Sal Khan considers two different ways to think about why a number raised to the zero power equals one: 1) if 2^3 = 1x2x2x2, then 2^0 = 1 times zero twos, which equals 1. We can use the product rule for exponents no matter what the exponent looks like, as long as the base is the same. Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next). For now consider 00 to be undefined. But what happens when you raise a number to the 0 0 power like this? The product rule for exponents:For any number, The quotient rule for exponents:For any non-zero number. Raise a number to the power of 1 means you have one of that number, raise to the power of 2 means you have two of the number multiplied together, power 3 means three of the number multiplied. An exponential number can be written as a n, where a = base and n = exponent. This leads to another rule for exponentsthe Power Rule for Exponents. Do not simplify further. We and our partners use cookies to Store and/or access information on a device. In the following video, we provide more examples of how to find the power of a product. Therefore, in the expression [latex]xy^{4}[/latex],only the y is affected by the 4. [latex]3^x=3^4=3\cdot3\cdot3\cdot3=81[/latex], [latex]3^{x}=81[/latex] when [latex]x=4[/latex]. For any nonzero real number [latex]a[/latex], the zero exponent rule of exponents states that. [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}=\left({x}^{2}\cdot {x}^{5}\right)\cdot {x}^{3}=\left({x}^{2+5}\right)\cdot {x}^{3}={x}^{7}\cdot {x}^{3}={x}^{7+3}={x}^{10}[/latex], [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[/latex], [latex]\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[/latex], [latex]\Large\begin{array}{ccc}\hfill \frac{{h}^{3}}{{h}^{5}}& =& \frac{h\cdot h\cdot h}{h\cdot h\cdot h\cdot h\cdot h}\hfill \\ & =& \frac{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}}{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}\cdot h\cdot h}\hfill \\ & =& \frac{1}{h\cdot h}\hfill \\ & =& \frac{1}{{h}^{2}}\hfill \end{array}[/latex], [latex]\Large\begin{array}{ccc}\hfill \frac{{h}^{3}}{{h}^{5}}& =& {h}^{3 - 5}\hfill \\ & =& \text{ }{h}^{-2}\hfill \end{array}[/latex], [latex]\begin{array}{ccc}{a}^{-n}=\frac{1}{{a}^{n}}& \text{and}& {a}^{n}=\frac{1}{{a}^{-n}}\end{array}[/latex], [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex], [latex]\begin{array}{ccc}\hfill {\left(pq\right)}^{3}& =& \stackrel{3\text{ factors}}{{\left(pq\right)\cdot \left(pq\right)\cdot \left(pq\right)}}\hfill \\ & =& p\cdot q\cdot p\cdot q\cdot p\cdot q\hfill \\ & =& \stackrel{3\text{ factors}}{{p\cdot p\cdot p}}\cdot \stackrel{3\text{ factors}}{{q\cdot q\cdot q}}\hfill \\ & =& {p}^{3}\cdot {q}^{3}\hfill \end{array}[/latex], [latex]{\left(ab\right)}^{n}={a}^{n}{b}^{n}[/latex], [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}=\frac{{f}^{14}}{{e}^{14}}[/latex], [latex]\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}[/latex], [latex]\normalsize\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{\left({f}^{2}\right)}^{7}}{{\left({e}^{2}\right)}^{7}}\hfill \\ & =& \frac{{f}^{2\cdot 7}}{{e}^{2\cdot 7}}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}[/latex], [latex]{\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}[/latex], http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface, http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface, [latex]\left(3a\right)^{7}\cdot\left(3a\right)^{10} [/latex], [latex]\left(\left(3a\right)^{7}\right)^{10} [/latex], [latex]\left(3a\right)^{7\cdot10} [/latex], Define and identify notation and terminology for exponents, Identify the components of a term containing integer exponents, Evaluate expressions containing integer exponents, Use the product rule to multiply exponential expressions, Use the quotient rule to divide exponential expressions, Use the power rule to simplify expressions with exponents raised to powers, Define and use the negative exponent rule, Find the Power of a Product and a Quotient, Simplify an expression with a product raised to a power, Simplify an expression with a quotient raised to a power, Combine all the exponents rules to simplify expressions, [latex]{\left(\frac{1}{2}\right)}^{3}[/latex], [latex]\left(5x\right)^{3}[/latex]if [latex]x=4[/latex], [latex]x^{3}[/latex] if [latex]x=4[/latex], [latex]3^x[/latex] if [latex]x = 4[/latex], [latex]\left(-3\right)^{5}\cdot \left(-3\right)[/latex], [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}[/latex], [latex]{t}^{5}\cdot {t}^{3}={t}^{5+3}={t}^{8}[/latex], [latex]{\left(-3\right)}^{5}\cdot \left(-3\right)={\left(-3\right)}^{5}\cdot {\left(-3\right)}^{1}={\left(-3\right)}^{5+1}={\left(-3\right)}^{6}[/latex], [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}[/latex], [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}[/latex], [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}={\left(-2\right)}^{14 - 9}={\left(-2\right)}^{5}[/latex], [latex]\frac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[/latex], [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}={\left(z\sqrt{2}\right)}^{5 - 1}={\left(z\sqrt{2}\right)}^{4}[/latex], [latex]{\left({x}^{2}\right)}^{7}[/latex], [latex]{\left({\left(2t\right)}^{5}\right)}^{3}[/latex], [latex]{\left({\left(-3\right)}^{5}\right)}^{11}[/latex], [latex]{\left({x}^{2}\right)}^{7}={x}^{2\cdot 7}={x}^{14}[/latex], [latex]{\left({\left(2t\right)}^{5}\right)}^{3}={\left(2t\right)}^{5\cdot 3}={\left(2t\right)}^{15}[/latex], [latex]{\left({\left(-3\right)}^{5}\right)}^{11}={\left(-3\right)}^{5\cdot 11}={\left(-3\right)}^{55}[/latex], [latex]\Large\frac{{c}^{3}}{{c}^{3}}[/latex], [latex]\Large\frac{-3{x}^{5}}{{x}^{5}}[/latex], [latex]\Large\frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}[/latex], [latex]\Large\frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}[/latex], [latex]\Large\begin{array}\text{ }\frac{c^{3}}{c^{3}} \hfill& =c^{3-3} \\ \hfill& =c^{0} \\ \hfill& =1\end{array}[/latex], [latex]\Large\begin{array}{ccc}\hfill \frac{-3{x}^{5}}{{x}^{5}}& =& -3\cdot \frac{{x}^{5}}{{x}^{5}}\hfill \\ & =& -3\cdot {x}^{5 - 5}\hfill \\ & =& -3\cdot {x}^{0}\hfill \\ & =& -3\cdot 1\hfill \\ & =& -3\hfill \end{array}[/latex], [latex]\large\begin{array}{cccc}\hfill \frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}& =& \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{1+3}}\hfill &\text{Use the product rule in the denominator}.\hfill \\ & =&\frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{4}}\hfill & \text{Simplify}.\hfill \\ & =&{\left({j}^{2}k\right)}^{4 - 4}\hfill &\text{Use the quotient rule}.\hfill \\ & =& {\left({j}^{2}k\right)}^{0}\hfill&\text{Simplify}.\hfill \\ & =& 1& \end{array}[/latex], [latex]\large\begin{array}{cccc}\hfill \frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}& =&5{\left(r{s}^{2}\right)}^{2 - 2}\hfill &\text{Use the quotient rule}.\hfill \\ & =& 5{\left(r{s}^{2}\right)}^{0}\hfill &\text{Simplify}.\hfill \\ & =& 5\cdot 1\hfill &\text{Use the zero exponent rule}.\hfill \\ & =& 5\hfill &\text{Simplify}.\hfill \end{array}[/latex], [latex]\Large\frac{{(2b) }^{3}}{{(2b) }^{10}}[/latex], [latex]\Large\frac{{z}^{2}\cdot z}{{z}^{4}}[/latex], [latex]\Large\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}[/latex], [latex]\Large\frac{{(2b) }^{3}}{{(2b)}^{10}}={(2b)}^{3 - 10}={(2b) }^{-7}=\frac{1}{{(2b)}^{7}}[/latex], [latex]\Large\frac{{z}^{2}\cdot z}{{z}^{4}}=\frac{{z}^{2+1}}{{z}^{4}}=\frac{{z}^{3}}{{z}^{4}}={z}^{3 - 4}={z}^{-1}=\frac{1}{z}[/latex], [latex]\Large\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}={\left(-5{t}^{3}\right)}^{4 - 8}={\left(-5{t}^{3}\right)}^{-4}=\frac{1}{{\left(-5{t}^{3}\right)}^{4}}[/latex], [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}[/latex], [latex]\frac{-7z}{{\left(-7z\right)}^{5}}[/latex], [latex]{b}^{2}\cdot {b}^{-8}={b}^{2 - 8}={b}^{-6}=\frac{1}{{b}^{6}}[/latex], [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}={\left(-x\right)}^{5 - 5}={\left(-x\right)}^{0}=1[/latex], [latex]\frac{-7z}{{\left(-7z\right)}^{5}}=\frac{{\left(-7z\right)}^{1}}{{\left(-7z\right)}^{5}}={\left(-7z\right)}^{1 - 5}={\left(-7z\right)}^{-4}=\frac{1}{{\left(-7z\right)}^{4}}[/latex], [latex]{\left(a{b}^{2}\right)}^{3}[/latex], [latex]{\left(2^a{t}\right)}^{15}[/latex], [latex]{\left(-2{w}^{3}\right)}^{3}[/latex], [latex]\frac{1}{{\left(-7z\right)}^{4}}[/latex], [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}[/latex], [latex]{\left(a{b}^{2}\right)}^{3}={\left(a\right)}^{3}\cdot {\left({b}^{2}\right)}^{3}={a}^{1\cdot 3}\cdot {b}^{2\cdot 3}={a}^{3}{b}^{6}[/latex], [latex]{\left(2^a{t}\right)}^{15}={\left(2^a\right)}^{15}\cdot {\left(t\right)}^{15}={2}^{a\cdot15}\cdot{t}^{15}=2^{15a}\cdot{t}^{15}[/latex], [latex]{\left(-2{w}^{3}\right)}^{3}={\left(-2\right)}^{3}\cdot {\left({w}^{3}\right)}^{3}=-8\cdot {w}^{3\cdot 3}=-8{w}^{9}[/latex], [latex]\frac{1}{{\left(-7z\right)}^{4}}=\frac{1}{{\left(-7\right)}^{4}\cdot {\left(z\right)}^{4}}=\frac{1}{2,401{z}^{4}}[/latex], [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}={\left({e}^{-2}\right)}^{7}\cdot {\left({f}^{2}\right)}^{7}={e}^{-2\cdot 7}\cdot {f}^{2\cdot 7}={e}^{-14}{f}^{14}=\frac{{f}^{14}}{{e}^{14}}[/latex], [latex]{\left(\frac{4}{{z}^{11}}\right)}^{3}[/latex], [latex]{\left(\frac{p}{{q}^{3}}\right)}^{6}[/latex], [latex]{\left(\frac{-1}{{t}^{2}}\right)}^{27}[/latex], [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}[/latex], [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}[/latex], [latex]\Large{\left(\frac{4}{{z}^{11}}\right)}^{3}=\frac{{\left(4\right)}^{3}}{{\left({z}^{11}\right)}^{3}}=\frac{64}{{z}^{11\cdot 3}}=\frac{64}{{z}^{33}}[/latex], [latex]\Large{\left(\frac{p}{{q}^{3}}\right)}^{6}=\frac{{\left(p\right)}^{6}}{{\left({q}^{3}\right)}^{6}}=\frac{{p}^{1\cdot 6}}{{q}^{3\cdot 6}}=\frac{{p}^{6}}{{q}^{18}}[/latex], [latex]\Large{\left(\frac{-1}{{t}^{2}}\right)}^{27}=\frac{{\left(-1\right)}^{27}}{{\left({t}^{2}\right)}^{27}}=\frac{-1}{{t}^{2\cdot 27}}=\frac{-1}{{t}^{54}}=-\frac{1}{{t}^{54}}[/latex], [latex]\Large{\left({j}^{3}{k}^{-2}\right)}^{4}={\left(\frac{{j}^{3}}{{k}^{2}}\right)}^{4}=\frac{{\left({j}^{3}\right)}^{4}}{{\left({k}^{2}\right)}^{4}}=\frac{{j}^{3\cdot 4}}{{k}^{2\cdot 4}}=\frac{{j}^{12}}{{k}^{8}}[/latex], [latex]\Large{\left({m}^{-2}{n}^{-2}\right)}^{3}={\left(\frac{1}{{m}^{2}{n}^{2}}\right)}^{3}=\frac{{\left(1\right)}^{3}}{{\left({m}^{2}{n}^{2}\right)}^{3}}=\frac{1}{{\left({m}^{2}\right)}^{3}{\left({n}^{2}\right)}^{3}}=\frac{1}{{m}^{2\cdot 3}\cdot {n}^{2\cdot 3}}=\frac{1}{{m}^{6}{n}^{6}}[/latex]. The exponent is another name for the power of a number. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and 2. Note that there are several different ways we can manipulate the term Bx/y to do some algebraic gymnastics: A decimal exponent can often be converted to a fraction exponent. Evaluate. If a nonzero base is raised to a negative exponent, then the result is the reciprocal with a positive exponent. Evaluate the following expressions for the given value. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Dont forget to subscribe to our YouTube channel & get updates on new math videos! In the following video you will see more examples of using the power rule to simplify expressions with exponents. a n times a is the base and n is the exponent. Finally, this expression can be rewritten as [latex]4^{3}[/latex]using exponential notation. Sort by: Top Voted Micah Kunkle 9 years ago Why is a number. Be careful that you subtract the exponent in the denominator from the exponent in the numerator. When [latex]a

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