Important rules to simplify radical expressions and expressions with exponents are presented along with examples. Questions with answers are at the bottom of the page.
| Rules | Examples |
| \( 0^ 0\) 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 even integer | \( (-1)^6 = 1 \) |
| \( (-1)^m = - 1 \) if \( m \) is odd integer | \( (-1)^9 = - 1 \) |
| \( x^m x^n = x^{m+n} \) | \( 2^3 2^4 = 2^{3 + 4} = 2^7 \) |
| \( \dfrac{x^m}{x^n} = x^{m-n} \) | \( \dfrac{7^5}{7^2} = 7^{5-2} = 7^3 \) |
| \( (x^m)^n = x^{m \times n} \) | \( (4^5)^2= 4^{5 \times 2} = 4^{10} \) |
| \( (x \; y)^m = x^m \; y^m \) | \( (3 \; b)^2 = 3^2 \; b^2 = 9 b^2 \) |
| \( \left ( \dfrac{x}{y} \right )^m = \dfrac{x^m}{y^m} \; , y \ne 0 \) | \( \left ( \dfrac{4}{b} \right )^2 = \dfrac{4^2}{b^2} = \dfrac{16}{b^2} \) |
| \( (-x)^m = (-1)^m \; x^m \) | \( (-3)^4 = (-1)^4 (3^4) = 1 \times 81 = 81 \) |
| \( \left (\dfrac{x}{y} \right)^{-m} = \left (\dfrac{y}{x} \right)^{m} \; , \; x \ne 0 \; , y \; \ne 0 \) | \( \left (\dfrac{4}{3} \right)^{-2} = \left (\dfrac{3}{4} \right)^{2} \) |
| \( \dfrac{1}{y^{-m}} = y^m \; , \; y \ne 0 \) | \( \dfrac{1}{8^{-2}} = 8^2 \) |
| \( | x^m| = | x |^m \) | \( | (- 5)^4 | = |-5| ^4 = 5^4 = 625 \) |
The principal \( n^{nt} \) root of a number \( x \) has the same sign as \( x \).
1) The square (second) root of \( 4 \) is \( 2 \) (Note: - 2 is also a root but it is not the principal because it has opposite site to 4)
2) The cube (third) root of \( 8 \) is \( 2 \)
4) The cube (third) root of \( - 8 \) is \( - 2 \)
Special symbols called radicals are used to indicate the principal root of a number.
\[ \huge \color{red}{ y = \sqrt[n]{x} } \] \( n \) is the index, \( x \) is the radicand. For the square root (n = 2), we dot write the index.
| Rules | Examples |
| \( \sqrt[n]{x^m} = (\sqrt[n]{x})^m \) | \(\sqrt[3]{27^2} = (\sqrt[3]{27})^2 = 3^2 \) |
| \( x^{m/n} = (\sqrt[n]{x})^m = \sqrt[n]{x^m} \) | \(4^{3/2} = (\sqrt{4})^3 = 2^3 \) |
| \( \sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{x y} \) | \(\sqrt[5]{16} \cdot \sqrt[5]{2} = \sqrt[5]{16 \times 2} = \sqrt[5]{32} = 2 \) |
| \( \dfrac{\sqrt[n]{x}}{\sqrt[n]{y}} = \sqrt[n]{\dfrac{x}{y}} \) | \(\dfrac{\sqrt[3]{-40}}{\sqrt[3]{5}} = \sqrt[3]{\dfrac{-40}{5}} = \sqrt[3]{-8} = - 2 \) |
| \( (\sqrt[m]{x})^m = x \) | \((\sqrt[3]{ - 2})^3 = - 2\) |
| \( \sqrt[m]{x^m} = | x | \;\; \text{if m is even} \) | \(\sqrt[4]{(- 2)^4} = | - 2| = 2 \) |
| \( \sqrt[m]{x^m} = x \;\; \text{if m is odd} \) | \(\sqrt[5]{ (- 2)^5} = - 2 \) |