Comparing Radicals

To compare two or more radicals, rewrite them with a common index and then compare their radicands.

An illustrative example

Consider the following two radicals

$$ \sqrt[6]{5} $$

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

To determine which radical is larger, begin by expressing both radicals with a common index.

The least common multiple of the indices 6 and 4 is 12.

By the invariance property of radicals, multiply both the index and the exponent of the first radical by 2.

Then multiply both the index and the exponent of the second radical by 3.

$$ \sqrt[6]{5} = \sqrt[6 \cdot 2]{(5)^2} = \sqrt[12]{5^2} $$

$$ \sqrt[4]{3} = \sqrt[4 \cdot 3]{(3)^3} = \sqrt[12]{3^3} $$

The two radicals now share the same index.

$$ \sqrt[12]{5^2} $$

$$ \sqrt[12]{3^3} $$

Next, evaluate the exponents inside the radicals.

$$ \sqrt[12]{25} $$

$$ \sqrt[12]{27} $$

The second radical has the greater radicand.

$$ 25 < 27 $$

Since the radicals have the same index, the one with the larger radicand is greater.

$$ \sqrt[12]{25} < \sqrt[12]{27} $$

Verification. To confirm the result, raise both radicals to the twelfth power. $$ ( \sqrt[12]{25} )^{12} = 25 $$ $$ ( \sqrt[12]{27} )^{12} = 27 $$ This eliminates the radical sign and returns the original radicands. Because 27 is greater than 25, it follows that the second radical is greater than the first.

This method applies generally when comparing radicals with different indices.