Exponent Rules Explained

This post will help you understand the rules of exponents so they are easier to remember. You will also learn to apply them for simplifying expressions with exponents. Let’s take a look at the most commonly used exponent rules explained.

Definition of exponent a^n=a*a*a*...*a (n times)
Product rule of exponents explained with proof a^m*a^n=a^(m+n)

Product rule

Quotient rule of exponents explained with proof a^m/a^n=a^m-n

Quotient rule

Zero exponent rule explained with proof a^0=1

Zero exponent rule

Negative exponent rule explained with proof a^-n=1/a^n

Negative exponent rule

Rational exponent rule explained with proof a^(m/n)=nroot(a^m)

Rational exponent rule

Power of power rule explained with proof (a^m)^n=a^(m*n)

Power of a power rule

Power of product rule explained with proof (ab)^m=a^m*b^m

Power of a product rule

Power of quotient rule explained with proof (a/b)^m=a^m/b^m

Power of a quotient rule

Exponent rules are an essential tool for simplifying exponential expressions and solving exponential equations. Check out the tips and examples below to see these rules in action. And keep scrolling to find practice problems with answer key.

Tips for using exponent rules. Expressions with negative exponents.
Tips for using exponent rules. Expressions with parentheses.

As always, practice makes better. So try out the problems below to practice and improve your skills. Find the answer key at the end of this page to check your work. If you get stuck, review the rules and examples and give it another try. Or watch this video for a step-by-step example.

Simplify the expression completely.

Expression with exponents practice question with answer key (81x)^(3/4)

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Expression with exponents practice question with answer key (3x)^(1/3)*(9x^2)^(1/3)

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Expression with exponents practice question with answer key (3x^(-1/2)*y^5)/(2^2x^-2y)

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Expression with exponents practice question with answer key x^2/y^5*(2y)/(3x^-1)

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Expression with exponents simplified using exponent rules step by step example (2x^0*x^3)/(2x)^-1

Example 1

Expression with exponents simplified using exponent rules step by step example 2x^3(4x)^2

Example 2

Expression with exponents simplified using exponent rules step by step example (x^2(xy^3)^2))/(xy^6)

Example 3

Expression with exponents simplified using exponent rules step by step example x^2/x^5

Example 4

Tips for using exponent rules. Expressions with exponent 1.
Expression with exponents simplified using exponent rules step by step example (2^5x^-3*(2y)^-2)/(3x^-5*y^-4)

Example 5

Expression with exponents simplified using exponent rules step by step example (8x)^(2/3)

Example 6

Expression with exponents practice question with answer key ((5x^-1*x^(2/3))/(y^5*x^2))^0

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Expression with exponents practice question with answer key (4x^(1/2)*y^3*(3x)^-1y^2)^2

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Exponents may seem intimidating at first, but with a solid understanding of their rules and applications you can master this important topic. Furthermore, repeated exposure to these rules can help increase your mathematical intuition.

Answer key

Expressions with exponents practice questions answer key.

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