# Reduce the index on a radical and multiply or divide radicals of different index

Knowing that a radical has the same properties as exponents (written as a ratio) allows us to manipulate radicals in new ways. One thing we are allowed to do is reduce, not just the radicand, but the index as well. This is shown in the following example.

Example 1:

Reduce $\sqrt[8]{x^6y^2}$

Rewrite as rational exponent

$(x^6y^2)^{\frac{1}{8}}$

Multiply exponents

$x^{\frac{6}{8}}y^{\frac{2}{8}}$

Reduce each fraction

$x^{\frac{3}{4}}y^{\frac{1}{4}}$

All exponents have denominator of 4, this is our new index. And our solution is

$\sqrt[4]{x^3y}$

What we have done is reduced our index by dividing the index and all the exponents by the same number (2 in the previous example). If we notice a common factor in the index and all the exponents on every factor we can reduce by dividing by that common factor. This is shown in the next example.

Example 2:

Reduce $\sqrt[24]{a^6b^9c^{15}}$

Index and all exponents are divisible by 3, and our solution is

Reduce $\sqrt[8]{a^2b^3c^{5}}$

We can use the same process when there are coefficients in the problem. We will first write the coefficient as an exponential expression so we can divide the exponent by the common factor as well.

Example 3:

Reduce $\sqrt[9]{8m^6n^3}$

$\sqrt[9]{8m^6n^3}$

Write 8 as 23

$\sqrt[9]{2^3m^6n^3}$

Index and all exponents are divisible by 3

And our solution is

$\sqrt[3]{2m^2n}$.

We can use a very similar idea to also multiply radicals where the index does not match. First we will consider an example using rational exponents, then identify the pattern we can use.

Example 4:

Reduce $\sqrt[3]{ab^2}\sqrt[4]{a^2b}$

$\sqrt[3]{ab^2}\sqrt[4]{a^2b}$

Rewrite as rational exponents

$(ab^2)^{\frac{1}{3}}(a^2b)^{\frac{1}{4}}$

Multiply exponents

$a^{\frac{1}{3}}b^{\frac{2}{3}}a^{\frac{2}{4}}b^{\frac{1}{4}}$

To have one radical need a common denominator, 12

$a^{\frac{4}{12}}b^{\frac{8}{12}}a^{\frac{6}{12}}b^{\frac{3}{12}}$

Write under a single radical with common index, 12

$\sqrt[12]{a^4b^8a^6b^3}$

$\sqrt[12]{a^{10}b^{11}}$ (our solution).

To combine the radicals we need a common index (just like the common denominator). We will get a common index by multiplying each index and exponent by an integer that will allow us to build up to that desired index. This process is shown in the next example.

Example 5:

Reduce $\sqrt[4]{a^2b^3}\sqrt[6]{a^2b}$

$\sqrt[12]{a^6b^9a^4b^2}$
$\sqrt[12]{a^{10}b^{11}}$