Roots are the inverse operation of exponentiation. They are used to find a value that, when raised to a certain power, gives the original number. These rules help simplify radical expressions and are especially useful in algebra and calculus.
Raising a root to its corresponding power gives the original number:
\[ \left( \sqrt[n]{a} \right)^n = a \]
where:
The exponent can be moved outside as a power of the root:
\[ \sqrt[n]{a^m} = \left( \sqrt[n]{a} \right)^m \]
The root of a product equals the product of the roots:
\[ \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} \]
The root of a quotient equals the quotient of the roots:
\[ \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad (b \neq 0) \]
Nesting roots multiplies their indices:
\[ \sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a} \]
When two powers are multiplied inside a root:
\[ \sqrt[n]{a^m \cdot b^n} = \sqrt[n]{a^m} \cdot \sqrt[n]{b^n} \]
Combining different roots under division:
\[ \frac{\sqrt[n]{a}}{\sqrt[m]{b}} = \sqrt[\frac{nm}{mn}]{\frac{a^m}{b^n}} \]
To remove a root from the denominator:
\[ \frac{x}{\sqrt[n]{a}} = \frac{x \cdot \sqrt[n]{a^{n-1}}}{a} \quad (a \neq 0) \]
Use the conjugate to rationalize:
\[ \frac{x}{\sqrt{a} + \sqrt{b}} = \frac{x(\sqrt{a} - \sqrt{b})}{a - b} \]