Powers of Radicals

The m-th power of a radical is a radical with the same index, where the radicand is raised to the m-th power. $$ (\sqrt[n]{a} )^m = \sqrt[n]{a^m} $$ Here, n and m are natural numbers greater than zero. The radicand a is a real number, subject to the condition a ≥ 0 when n is even.

A Practical Example

Consider the expression

$$ ( \sqrt[3]{2^2} )^4 $$

Raise the radicand to the fourth power

$$ \sqrt[3]{(2^2)^4} $$

Next, apply the laws of exponents

$$ \sqrt[3]{2^{2 \cdot 4}} $$

This gives a radical equivalent to the original expression

$$ \sqrt[3]{2^8} $$

Proof

Begin with the identity

$$ (\sqrt[n]{a} )^m = \sqrt[n]{a^m} $$

Raise both sides to the n-th power

$$ [ (\sqrt[n]{a} )^m ]^n = [ \sqrt[n]{a^m} ]^n $$

Simplify the right-hand side

$$ [ (\sqrt[n]{a} )^m ]^n = [ \sqrt[\not{n}]{a^m} ]^{\not{n}} $$

$$ [ (\sqrt[n]{a} )^m ]^n = a^m $$

Apply the laws of exponents to the left-hand side

$$ (\sqrt[n]{a} )^{m \cdot n} = a^m $$

Now simplify the radical by dividing both the index of the root and the exponent of the radicand by n

$$ (\sqrt[\frac{n}{n}]{a} )^{\frac{m \cdot n}{n}} = a^m $$

$$ a^m = a^m $$

This establishes that the two sides of the original equation are equal.

The power rule for radicals is therefore proven.

And so on.