Important rules to simplify radical expressions and expressions with exponents are presented below along with examples. Interactive practice questions with step-by-step answers are provided at the bottom of the page.
| Rule | Example |
|---|---|
| \( 0^0 \) is undefined | |
| \( 0^m = 0 \) , \( m \gt 0 \) | \( 0^{10} = 0 \) |
| \( x^0 = 1 \; , \; x \ne 0 \) | \( 21^0 = 1 \) |
| \( 1^m = 1 \) | \( 1^{12} = 1 \) |
| \( (-1)^m = 1 \) if \( m \) is an even integer | \( (-1)^6 = 1 \) |
| \( (-1)^m = -1 \) if \( m \) is an odd integer | \( (-1)^9 = -1 \) |
| \( x^m x^n = x^{m+n} \) | \( 2^3 2^4 = 2^{3 + 4} = 2^7 \) |
| \( \dfrac{x^m}{x^n} = x^{m-n} \) | \( \dfrac{7^5}{7^2} = 7^{5-2} = 7^3 \) |
| \( (x^m)^n = x^{m \times n} \) | \( (4^5)^2= 4^{5 \times 2} = 4^{10} \) |
| \( (x \; y)^m = x^m \; y^m \) | \( (3 \; b)^2 = 3^2 \; b^2 = 9 b^2 \) |
| \( \left ( \dfrac{x}{y} \right )^m = \dfrac{x^m}{y^m} \; , y \ne 0 \) | \( \left ( \dfrac{4}{b} \right )^2 = \dfrac{4^2}{b^2} = \dfrac{16}{b^2} \) |
| \( (-x)^m = (-1)^m \; x^m \) | \( (-3)^4 = (-1)^4 (3^4) = 1 \times 81 = 81 \) |
| \( \left (\dfrac{x}{y} \right)^{-m} = \left (\dfrac{y}{x} \right)^{m} \; , \; x \ne 0 \; , y \ne 0 \) | \( \left (\dfrac{4}{3} \right)^{-2} = \left (\dfrac{3}{4} \right)^{2} \) |
| \( \dfrac{1}{y^{-m}} = y^m \; , \; y \ne 0 \) | \( \dfrac{1}{8^{-2}} = 8^2 \) |
| \( | x^m| = | x |^m \) | \( | (- 5)^4 | = |-5|^4 = 5^4 = 625 \) |
If \( x = y^n \) , then \( x \) is the \( n^{th} \) root of \( y \).
The principal \( n^{th} \) root of a number \( x \) has the same sign as \( x \).
Special symbols called radicals are used to indicate the principal root of a number.
\[ y = \sqrt[n]{x} \]
\( n \) is the index, and \( x \) is the radicand. For the square root (\( n = 2 \)), we do not write the index.
| Rule | Example |
|---|---|
| \( \sqrt[n]{x^m} = (\sqrt[n]{x})^m \) | \( \sqrt[3]{27^2} = (\sqrt[3]{27})^2 = 3^2 \) |
| \( x^{m/n} = (\sqrt[n]{x})^m = \sqrt[n]{x^m} \) | \( 4^{3/2} = (\sqrt{4})^3 = 2^3 \) |
| \( \sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{x y} \) | \( \sqrt[5]{16} \cdot \sqrt[5]{2} = \sqrt[5]{16 \times 2} = \sqrt[5]{32} = 2 \) |
| \( \dfrac{\sqrt[n]{x}}{\sqrt[n]{y}} = \sqrt[n]{\dfrac{x}{y}} \) | \( \dfrac{\sqrt[3]{-40}}{\sqrt[3]{5}} = \sqrt[3]{\dfrac{-40}{5}} = \sqrt[3]{-8} = -2 \) |
| \( (\sqrt[m]{x})^m = x \) | \( (\sqrt[3]{ - 2})^3 = - 2 \) |
| \( \sqrt[m]{x^m} = | x | \;\; \text{if m is even} \) | \( \sqrt[4]{(- 2)^4} = | - 2| = 2 \) |
| \( \sqrt[m]{x^m} = x \;\; \text{if m is odd} \) | \( \sqrt[5]{ (- 2)^5} = - 2 \) |
Use the rules listed above to simplify the following expressions and rewrite them with positive exponents. Note that sometimes you need to use more than one rule to simplify a given expression.
\( (-1)^{125} = -1 \)
\( 2^5 \; 2^{-2} = 2^3 = 8 \)
\( 9^3 / 9^5 = 1 / 9^2 = 1/81 \)
\( 0^3 = 0 \)
\( ( 2 / y)^5 = 2^5 / y^5 = 32 / y^5 \)
\( (- 3)^4 = 3^4 = 81 \)
\( (2 / 5)^{-1} = 5/2 \)
\( | - 2 |^4 = 2^4 = 16 \)
\( (-3)^0 = 1 \)
\( (- 1)^4 = 1 \)
\( (- 1)^{15} = -1 \)
\( (3^2)^3 = 3^6 = 729 \)
\( (- 4 x)^3 = (- 4)^3 x^3 = - 4^3 \; x^3 = - 64 x^3 \)
\( \sqrt[4]{16^3} = (\sqrt[4]{16})^3 = 2^3 = 8 \)
\( 27^{5/3} = (\sqrt[3]{27})^5 = 3^5 = 243 \)
\( \sqrt[3]{32} \cdot \sqrt[3]{2} = \sqrt[3]{64} = 4 \)
\( \dfrac{\sqrt{160}}{\sqrt{40}} = \sqrt{\dfrac{160}{40}} = \sqrt{4} = 2 \)
\( (\sqrt[6]{ 3})^6 = 3^{(6/6)} = 3^1 = 3 \)
\( \sqrt[4]{ (- 7)^4 } = | - 7 | = 7 \)
\( \sqrt[5]{(- 9)^5} = - 9 \)