Rules for Radicals and Exponents

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 for Exponents.

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 \( (-1)^9 = - 1 \)
\( x^m x^n = x^{m+n} \) \( 2^3 2^4 = 2^{3 + 4} = 2^7 \)
\( \dfrac{x^m}{x^n} = x^{m-n} \) \( \dfrac{7^5}{7^2} = 7^{5-2} = 7^3 \)
\( (x^m)^n = x^{m \times n} \) \( (4^5)^2= 4^{5 \times 2} = 4^{10} \)
\( (x \; y)^m = x^m \; y^m \) \( (3 \; b)^2 = 3^2 \; b^2 = 9 b^2 \)
\( \left ( \dfrac{x}{y} \right )^m = \dfrac{x^m}{y^m} \; , y \ne 0 \) \( \left ( \dfrac{4}{b} \right )^2 = \dfrac{4^2}{b^2} = \dfrac{16}{b^2} \)
\( (-x)^m = (-1)^m \; x^m \) \( (-3)^4 = (-1)^4 (3^4) = 1 \times 81 = 81 \)
\( \left (\dfrac{x}{y} \right)^{-m} = \left (\dfrac{y}{x} \right)^{m} \; , \; x \ne 0 \; , y \; \ne 0 \) \( \left (\dfrac{4}{3} \right)^{-2} = \left (\dfrac{3}{4} \right)^{2} \)
\( \dfrac{1}{y^{-m}} = y^m \; , \; y \ne 0 \) \( \dfrac{1}{8^{-2}} = 8^2 \)
\( | x^m| = | x |^m \) \( | (- 5)^4 | = |-5| ^4 = 5^4 = 625 \)

What are Radicals?

If \( x = y^n \) , then \( x \) is the \( n^{th} \) root of \( y \).
The principal \( n^{nt} \) root of a number \( x \) has the same sign as \( x \).
Examples
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 for Radicals.

Rules Examples
\( \sqrt[n]{x^m} = (\sqrt[n]{x})^m \) \(\sqrt[3]{27^2} = (\sqrt[3]{27})^2 = 3^2 \)
\( x^{m/n} = (\sqrt[n]{x})^m = \sqrt[n]{x^m} \) \(4^{3/2} = (\sqrt{4})^3 = 2^3 \)
\( \sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{x y} \) \(\sqrt[5]{16} \cdot \sqrt[5]{2} = \sqrt[5]{16 \times 2} = \sqrt[5]{32} = 2 \)
\( \dfrac{\sqrt[n]{x}}{\sqrt[n]{y}} = \sqrt[n]{\dfrac{x}{y}} \) \(\dfrac{\sqrt[3]{-40}}{\sqrt[3]{5}} = \sqrt[3]{\dfrac{-40}{5}} = \sqrt[3]{-8} = - 2 \)
\( (\sqrt[m]{x})^m = x \) \((\sqrt[3]{ - 2})^3 = - 2\)
\( \sqrt[m]{x^m} = | x | \;\; \text{if m is even} \) \(\sqrt[4]{(- 2)^4} = | - 2| = 2 \)
\( \sqrt[m]{x^m} = x \;\; \text{if m is odd} \) \(\sqrt[5]{ (- 2)^5} = - 2 \)

Questions

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. \( (-1)^{125} \)

  2. \( 2^5 \; 2^{-2} \)

  3. \( 9^3 / 9^5 \)

  4. \( 0^3 \)

  5. \( ( 2 / y)^5 \)

  6. \( (- 3)^4 \)

  7. \( (2 / 5)^{-1} \)

  8. \( | - 2 |^4 \)

  9. \( (-3)^0 \)

  10. \( (- 1)^4 \)

  11. \( (- 1)^{15} \)

  12. \( (3^2)^3 \)

  13. \( (- 4 x)^3 \)

  14. \(\sqrt[4]{16^3} \)

  15. \(27^{5/3} \)

  16. \(\sqrt[3]{32} \cdot \sqrt[3]{2} \)

  17. \(\dfrac{\sqrt{160}}{\sqrt{40}} \)

  18. \((\sqrt[6]{ 3})^6 \)

  19. \(\sqrt[4]{ (- 7)^4 }\)

  20. \(\sqrt[5]{(- 9)^5}\)

Answers to Above Questions


  1. \( (-1)^{125} = -1 \)

  2. \( 2^5 \; 2^{-2} = 2^3 = 8 \)

  3. \( 9^3 / 9^5 = 1 / 9^2 = 1/81 \)

  4. \( 0^3 = 0 \)

  5. \( ( 2 / y)^5 = 2^5 / y^5 = 32 / y^5\)

  6. \( (- 3)^4 = 3^4 = 81\)

  7. \( (2 / 5)^{-1} = 5/2\)

  8. \( | - 2 |^4 = 2^4 = 16\)

  9. \( (-3)^0 = 1 \)

  10. \( (- 1)^4 = 1\)

  11. \( (- 1)^{15} = -1\)

  12. \( (3^2)^3 = 3^6 = 729 \)

  13. \( (- 4 x)^3 = (- 4)^3 x^3 = - 4^3 \; x^3 = - 64 x^3 \)

  14. \(\sqrt[4]{16^3} = (\sqrt[4]{16})^3 = 2^3 = 8 \)

  15. \(27^{5/3} = (\sqrt[3]{27})^5 = 3^5 \)

  16. \(\sqrt[3]{32} \cdot \sqrt[3]{2} = \sqrt[3]{64} = 4 \)

  17. \(\dfrac{\sqrt{160}}{\sqrt{40}} = \sqrt{\dfrac{160}{40}} = \sqrt{4} = 2\)

  18. \((\sqrt[6]{ 3})^6 = 3^{(6/6)} = 3^1 = 3 \)

  19. \(\sqrt[4]{ (- 7)^4 } = | - 7 | = 7 \)

  20. \(\sqrt[5]{(- 9)^5} = - 9\)

More References and links

Simplify Radical Expressions
What are Exponents in Maths
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