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.
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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  \(\displaystyle{ \frac{x^n}{x^m} = x^{nm} }\) 
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.
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.
video by Mario's Math Tutoring 

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

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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.
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.
\( 52(x^{3}x^4)^0 \)
Problem Statement 

Simplify \( 52(x^{3}x^4)^0 \) using the exponent rules, giving your answer in exact form.
Final Answer 

\( 3 \)
Problem Statement 

Simplify \( 52(x^{3}x^4)^0 \) 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.
video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 \( (5/3) (x/y)^11 \)
video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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 

video by The Organic Chemistry Tutor 

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.