Laws of Exponents and Radicals

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index law
radical
exponent
power

Laws of Exponents (Index Law)
1. $ x^n = x \cdot x \cdot x ... \, (n \text{ factors}) $

2. $ x^m \cdot x^n = x^{m + n} $

3. $ (x^m)^n = x^{mn} $

4. $ (xyz)^n = x^n \, y^n \, z^n $

5. $ \dfrac{x^m}{x^n} = x^{m - n} $

6. $ \left( \dfrac{x}{y} \right)^n = \dfrac{x^n}{y^n} $

7. $ x^{-n} = \dfrac{1}{x^n} $   and   $ \dfrac{1}{x^{-n}} = x^n $

8. $ x^0 = 1 $,   provided   $ x \ne 0 $.

9. $ (x^m)^{1/n} = (x^{1/n})^m = x^{m/n} $

10. $ x^{m/n} = \sqrt[n]{x^m} $

11. If   $ x^m = x^n $,   then   $ m = n $   provided   $ x \ne 0 $.
 

Properties of Radicals
1. $ \sqrt[n]{x} = x^{1/n} $

2. $ \sqrt[n]{x^m} = \left( \sqrt[n]{x} \right)^m = x^{m/n} $

3. $ \sqrt[n]{x} \, \sqrt[n]{y} = \sqrt[n]{xy} $

4. $ \dfrac{\sqrt[n]{x}}{\sqrt[n]{y}} = \sqrt[n]{\dfrac{x}{y}} $

5. $ \sqrt[n]{x} \, \sqrt[m]{x} = \sqrt[mn]{x^{m + n}} $

6. $ \dfrac{\sqrt[n]{x}}{\sqrt[m]{x}} = \sqrt[mn]{x^{m - n}} $

7. $ \left( \sqrt[n]{x} \right)^n = x $
 

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