Nested Radicals

The m-th root of a radical with index n is a radical with the same radicand and a root index equal to the product of the two indices n*m $$ \sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a} $$

This identity also implies the commutative property of the indices.

$$ \sqrt[m]{\sqrt[n]{a}} = \sqrt[n]{\sqrt[m]{a}} $$

In other words, the order of the root indices can be interchanged.

An example

Consider the nested radical

$$ \sqrt[4]{\sqrt[3]{a^4}} $$

Multiplying the indices of the radicals gives

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

Hence

$$ \sqrt[12]{a^4} $$

Now simplify the index of the radical and the exponent of the radicand by dividing both by 4

$$ \sqrt[\frac{12}{4}]{a^\frac{4}{4}} $$

$$ \sqrt[3]{a} $$

Note. The expression can also be simplified more quickly by exchanging the indices of the radicals $$ \sqrt[4]{\sqrt[3]{a^4}} $$ $$ \sqrt[3]{\sqrt[4]{a^4}} $$ In this form the inner radical simplifies immediately because the root index and the exponent of the radicand are both equal to 4 $$ \sqrt[3]{\sqrt[\not{4}]{a^{\not{4}}}} $$ The final result is the same $$ \sqrt[3]{a} $$

Proof

A] The rule for nested radicals

To prove the identity, raise both sides of the equation to the power m*n

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

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

The radical on the right-hand side simplifies directly

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

Next apply the laws of exponents to the left-hand side

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

Simplify the exponent m with the root index m

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

Then simplify the exponent n with the root index n

$$ a = a $$

The two sides of the equation coincide.

This confirms the rule for nested radicals.

B] Exchanging the indices of the radicals

Consider the expression

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

Applying the rule for nested radicals gives

$$ \sqrt[m \cdot n]{a} $$

Since multiplication is commutative, m·n = n·m

$$ \sqrt[n \cdot m]{a} $$

Applying the rule for nested radicals in reverse yields

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

Therefore

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

This proves the commutative property of the indices of radicals.

And so on.