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17Calculus Precalculus - Exponent Rules

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In calculus you will be required to understand and be able to use the rules of exponents. Most instructors will not go over these rules before jumping into calculus. So make sure you know them well.

List of Rules

Let's start by going through these exponent rules.

1. The Product Rule

\( x^n \cdot x^m = x^{n+m} \)

2. The Quotient Rule

\(\displaystyle{ \frac{x^n}{x^m} = x^{n-m} }\)

3. The Power Rule

\( (x^n)^m = x^{n \cdot m} \)

4. Power of Zero

\( x^0 = 1 \)

5. Negative Exponents

\(\displaystyle{ x^{-n} = \frac{1}{x^n} }\)

6. Fractional Exponents

\( x^{1/n} = \sqrt[n]{x} \)

7. Distribute An Exponent Over a Product

\( (x \cdot y)^n = x^n y^n \)

8. Distribute An Exponent Over a Quotient

\(\displaystyle{ \left( \frac{x}{y} \right)^n = \frac{x^n}{y^n} }\)

In this video clip, the instructor explains these rules very well and includes some examples.

freeCodeCamp.org - College Algebra - Exponent Rules [10min-13secs]

video by freeCodeCamp.org

This instructor also does a great job explaining these rules. Before starting the practice problems, watch this video also to make sure you are ready.

Mario's Math Tutoring - Exponent Rules with Examples [6min-40secs]

Negative exponents can be one of the most difficult concepts when simplifying exponents. Here is a good video clip explaining negative exponents.

The Organic Chemistry Tutor - Negative Exponents Explained! [43secs]

Okay, let's practice these rules on these problems.

Practice Exponent Rules

Unless otherwise instructed, simplify these expressions using the exponent rules on this page. Give your answers in exact, simplified form.

\( x^{-3} \)

Problem Statement

Simplify \( x^{-3} \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 1/x^3 \)

Problem Statement

Simplify \( x^{-3} \) using the exponent rules, giving your answer in exact form.

Solution

3825 video

Final Answer

\( 1/x^3 \)

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\(\displaystyle{ \frac{1}{x^{-4}} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{1}{x^{-4}} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\( x^4 \)

Problem Statement

Simplify \(\displaystyle{ \frac{1}{x^{-4}} }\) using the exponent rules, giving your answer in exact form.

Solution

3826 video

Final Answer

\( x^4 \)

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Simplify \( - 3^2 \) and \( (-3)^2 \)

Problem Statement

Simplify \( - 3^2 \) and \( (-3)^2 \) using the exponent rules, giving your answer in exact form.

Final Answer

\( - 3^2 = -9 \) and \( (-3)^2 = 9 \)

Problem Statement

Simplify \( - 3^2 \) and \( (-3)^2 \) using the exponent rules, giving your answer in exact form.

Solution

3827 video

Final Answer

\( - 3^2 = -9 \) and \( (-3)^2 = 9 \)

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\( x^4 \cdot x^{-9} \)

Problem Statement

Simplify \( x^4 \cdot x^{-9} \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 1/x^5 \)

Problem Statement

Simplify \( x^4 \cdot x^{-9} \) using the exponent rules, giving your answer in exact form.

Solution

3828 video

Final Answer

\( 1/x^5 \)

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\(\displaystyle{ \left( \frac{3}{4} \right)^{-2} }\)

Problem Statement

Simplify \(\displaystyle{ \left( \frac{3}{4} \right)^{-2} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\(16/9\)

Problem Statement

Simplify \(\displaystyle{ \left( \frac{3}{4} \right)^{-2} }\) using the exponent rules, giving your answer in exact form.

Solution

3808 video

Final Answer

\(16/9\)

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\( (x^3)^4 \)

Problem Statement

Simplify \( (x^3)^4 \) using the exponent rules, giving your answer in exact form.

Final Answer

\( x^{12} \)

Problem Statement

Simplify \( (x^3)^4 \) using the exponent rules, giving your answer in exact form.

Solution

3824 video

Final Answer

\( x^{12} \)

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\( (3x^2)^3 \)

Problem Statement

Simplify \( (3x^2)^3 \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 27x^6 \)

Problem Statement

Simplify \( (3x^2)^3 \) using the exponent rules, giving your answer in exact form.

Solution

3832 video

Final Answer

\( 27x^6 \)

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\( (x^3)^{-4} \)

Problem Statement

Simplify \( (x^3)^{-4} \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 1/x^{12} \)

Problem Statement

Simplify \( (x^3)^{-4} \) using the exponent rules, giving your answer in exact form.

Solution

3809 video

Final Answer

\( 1/x^{12} \)

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\( x^4 \cdot x^5 \)

Problem Statement

Simplify \( x^4 \cdot x^5 \) using the exponent rules, giving your answer in exact form.

Final Answer

\( x^9 \)

Problem Statement

Simplify \( x^4 \cdot x^5 \) using the exponent rules, giving your answer in exact form.

Solution

3822 video

Final Answer

\( x^9 \)

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\( (x^3)^{-5} \)

Problem Statement

Simplify \( (x^3)^{-5} \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 1/x^{15} \)

Problem Statement

Simplify \( (x^3)^{-5} \) using the exponent rules, giving your answer in exact form.

Solution

3829 video

Final Answer

\( 1/x^{15} \)

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\( x^3 \cdot x^{-8} \)

Problem Statement

Simplify \( x^3 \cdot x^{-8} \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 1/x^5 \)

Problem Statement

Simplify \( x^3 \cdot x^{-8} \) using the exponent rules, giving your answer in exact form.

Solution

3810 video

Final Answer

\( 1/x^5 \)

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\( (4x^{-5})^2 \)

Problem Statement

Simplify \( (4x^{-5})^2 \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 16/x^{10} \)

Problem Statement

Simplify \( (4x^{-5})^2 \) using the exponent rules, giving your answer in exact form.

Solution

3811 video

Final Answer

\( 16/x^{10} \)

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\( \displaystyle{ \frac{x^7}{x^3} } \)

Problem Statement

Simplify \( \displaystyle{ \frac{x^7}{x^3} } \) using the exponent rules, giving your answer in exact form.

Final Answer

\( x^4 \)

Problem Statement

Simplify \( \displaystyle{ \frac{x^7}{x^3} } \) using the exponent rules, giving your answer in exact form.

Solution

3823 video

Final Answer

\( x^4 \)

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\(\displaystyle{ \frac{x^7}{x^{12}} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{x^7}{x^{12}} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\( 1/x^5 \)

Problem Statement

Simplify \(\displaystyle{ \frac{x^7}{x^{12}} }\) using the exponent rules, giving your answer in exact form.

Solution

3830 video

Final Answer

\( 1/x^5 \)

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\(\displaystyle{ \frac{y^5}{y^{-3}} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{y^5}{y^{-3}} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\( y^8 \)

Problem Statement

Simplify \(\displaystyle{ \frac{y^5}{y^{-3}} }\) using the exponent rules, giving your answer in exact form.

Solution

3812 video

Final Answer

\( y^8 \)

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\(\displaystyle{ \frac{x^{-8}}{x^{-11}} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{x^{-8}}{x^{-11}} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\( x^3 \)

Problem Statement

Simplify \(\displaystyle{ \frac{x^{-8}}{x^{-11}} }\) using the exponent rules, giving your answer in exact form.

Solution

3813 video

Final Answer

\( x^3 \)

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\(\displaystyle{ \frac{b^{-4}}{b^5} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{b^{-4}}{b^5} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\( 1/b^9 \)

Problem Statement

Simplify \(\displaystyle{ \frac{b^{-4}}{b^5} }\) using the exponent rules, giving your answer in exact form.

Solution

3819 video

Final Answer

\( 1/b^9 \)

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\(\displaystyle{ \frac{a^3}{a^7} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{a^3}{a^7} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\( 1/a^4 \)

Problem Statement

Simplify \(\displaystyle{ \frac{a^3}{a^7} }\) using the exponent rules, giving your answer in exact form.

Solution

3820 video

Final Answer

\( 1/a^4 \)

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Simplifying Using Exponent Rules

In this video clip, the instructor shows examples of how to use the exponent rules to simplify expressions.

freeCodeCamp.org - College Algebra - Simplifying [21min-32secs]

video by freeCodeCamp.org

Okay, let's practice simplifying on these problems.

Practice Simplifying

Unless otherwise instructed, simplify these expressions using the exponent rules on this page. Give your answers in exact, simplified form.

\( 5-2(x^{-3}x^4)^0 \)

Problem Statement

Simplify \( 5-2(x^{-3}x^4)^0 \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 3 \)

Problem Statement

Simplify \( 5-2(x^{-3}x^4)^0 \) using the exponent rules, giving your answer in exact form.

Solution

Comment: Although in general, his answer is correct, he is making an assumption that \(x \neq 0\). He doesn't need to state this since it is implied by the term \(x^{-3}\). If \(x = 0\) is part of the domain, this term would be \(0^{-3} = 1/0^3 = 1/0\). Remember in algebra, we never want the denominator to be zero. So, in this case the domain does not include zero. So he doesn't have to worry about the case \(0^0\) for this problem.

3814 video

Final Answer

\( 3 \)

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\( (-2x^3y^4)^2 \)

Problem Statement

Simplify \( (-2x^3y^4)^2 \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 4x^6y^8 \)

Problem Statement

Simplify \( (-2x^3y^4)^2 \) using the exponent rules, giving your answer in exact form.

Solution

3833 video

Final Answer

\( 4x^6y^8 \)

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\( (5x^3)(4x^7) \)

Problem Statement

Simplify \( (5x^3)(4x^7) \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 20x^{10} \)

Problem Statement

Simplify \( (5x^3)(4x^7) \) using the exponent rules, giving your answer in exact form.

Solution

3834 video

Final Answer

\( 20x^{10} \)

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\( (7x^6)(5x^4) \)

Problem Statement

Simplify \( (7x^6)(5x^4) \) using the exponent rules, giving your answer in exact form.

Final Answer

\( 35x^{10} \)

Problem Statement

Simplify \( (7x^6)(5x^4) \) using the exponent rules, giving your answer in exact form.

Solution

3835 video

Final Answer

\( 35x^{10} \)

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\( (4x^3y^2)(7x^{-4}y^3) \)

Problem Statement

Simplify \( (4x^3y^2)(7x^{-4}y^3) \) using the exponent rules, giving your answer in exact form.

Final Answer

\(\displaystyle{ \frac{28y^5}{x} }\)

Problem Statement

Simplify \( (4x^3y^2)(7x^{-4}y^3) \) using the exponent rules, giving your answer in exact form.

Solution

3836 video

Final Answer

\(\displaystyle{ \frac{28y^5}{x} }\)

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\( (3x^{-2}y^3)^2 (4x^5y^{-6})^3 \)

Problem Statement

Simplify \( (3x^{-2}y^3)^2 (4x^5y^{-6})^3 \) using the exponent rules, giving your answer in exact form.

Final Answer

\(\displaystyle{ \frac{576x^{11}}{y^{12}} }\)

Problem Statement

Simplify \( (3x^{-2}y^3)^2 (4x^5y^{-6})^3 \) using the exponent rules, giving your answer in exact form.

Solution

3816 video

Final Answer

\(\displaystyle{ \frac{576x^{11}}{y^{12}} }\)

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\(\displaystyle{ \frac{3a^{-4}}{b^{-6}c^3} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{3a^{-4}}{b^{-6}c^3} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\(\displaystyle{ \frac{3b^6}{a^4c^3} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{3a^{-4}}{b^{-6}c^3} }\) using the exponent rules, giving your answer in exact form.

Solution

3815 video

Final Answer

\(\displaystyle{ \frac{3b^6}{a^4c^3} }\)

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\(\displaystyle{ \frac{24x^7y^3}{8x^4y^{-12}} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{24x^7y^3}{8x^4y^{-12}} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\( 3x^3y^{15} \)

Problem Statement

Simplify \(\displaystyle{ \frac{24x^7y^3}{8x^4y^{-12}} }\) using the exponent rules, giving your answer in exact form.

Solution

3837 video

Final Answer

\( 3x^3y^{15} \)

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\(\displaystyle{ \frac{45x^4y^{-6}}{27x^{-7}y^5} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{45x^4y^{-6}}{27x^{-7}y^5} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\(\displaystyle{ \frac{5x^{11}}{3y^{11}} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{45x^4y^{-6}}{27x^{-7}y^5} }\) using the exponent rules, giving your answer in exact form.

Solution

Comments:
1. Notice in his solution, he started with the constants and simplified them first. This is a great way to make the problem seem not so complicated and we highly recommend this technique.
2. Although his answer is correct, you could also write the final answer as \( (5/3) (x/y)^11 \)

3817 video

Final Answer

\(\displaystyle{ \frac{5x^{11}}{3y^{11}} }\)

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\(\displaystyle{ \frac{18a^3b^{-4}c^{-6}}{30a^7b^5c^{-9}} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{18a^3b^{-4}c^{-6}}{30a^7b^5c^{-9}} }\) using the exponent rules, giving your answer in exact form.

Final Answer

\(\displaystyle{ \frac{3c^3}{5a^4b^9} }\)

Problem Statement

Simplify \(\displaystyle{ \frac{18a^3b^{-4}c^{-6}}{30a^7b^5c^{-9}} }\) using the exponent rules, giving your answer in exact form.

Solution

3818 video

Final Answer

\(\displaystyle{ \frac{3c^3}{5a^4b^9} }\)

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\( \displaystyle{ \left( \frac{35x^3y^5}{63x^{-4}y^7} \right)^2 } \)

Problem Statement

Simplify \(\displaystyle{ \left( \frac{35x^3y^5}{63x^{-4}y^7} \right)^2 }\) using the exponent rules, giving your answer in exact form.

Final Answer

\( \displaystyle{ \frac{25x^{14}}{81y^4} } \)

Problem Statement

Simplify \(\displaystyle{ \left( \frac{35x^3y^5}{63x^{-4}y^7} \right)^2 }\) using the exponent rules, giving your answer in exact form.

Solution

3831 video

Final Answer

\( \displaystyle{ \frac{25x^{14}}{81y^4} } \)

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\(\displaystyle{ \left( \frac{ 63x^{-3}y^4z^{-5} }{ 35x^8y^{-7}z^{-4} } \right)^2 }\)

Problem Statement

Simplify \(\displaystyle{ \left( \frac{ 63x^{-3}y^4z^{-5} }{ 35x^8y^{-7}z^{-4} } \right)^2 }\) using the exponent rules, giving your answer in exact form.

Final Answer

\(\displaystyle{ \frac{81y^{22}}{25x^{22}z^2} }\)

Problem Statement

Simplify \(\displaystyle{ \left( \frac{ 63x^{-3}y^4z^{-5} }{ 35x^8y^{-7}z^{-4} } \right)^2 }\) using the exponent rules, giving your answer in exact form.

Solution

3821 video

Final Answer

\(\displaystyle{ \frac{81y^{22}}{25x^{22}z^2} }\)

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Practice Instructions

Unless otherwise instructed, simplify these expressions using the exponent rules on this page. Give your answers in exact, simplified form.

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