Reducing Radicals to a Common Index

1. When dealing with two or more radicals that have different indices, rewrite them as equivalent radicals with a common index
2. Compute the least common multiple (LCM) of the radical indices, known as the common index.
3. Rewrite each radical as an equivalent radical whose index equals the common index.

This method yields equivalent radicals that share the same index.

A Practical Example

Consider the radicals

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

The domain restrictions are

$$ a,b \ge 0 $$

The radicals have different indices, namely 4 and 6.

$$ \sqrt[\color{red}4]{a^3} \cdot \sqrt[\color{red}6]{b} $$

To evaluate the product, first reduce the radicals to a common index.

Compute the least common multiple (LCM) of the indices.

$$ m.c.m.(4,6) = 12 $$

The common index is therefore 12.

By the invariance property of radicals, multiply both the index and the exponent of the radicand in the first radical by 3.

This produces an equivalent radical with index 12.

$$ \sqrt[4 \cdot \color{red}3]{(a^3)^\color{red}3} \cdot \sqrt[6]{b} $$

$$ \sqrt[12]{a^9} \cdot \sqrt[6]{b} $$

Apply the invariance property again, multiplying both the index and the exponent of the radicand in the second radical by 2.

This yields another equivalent radical with index 12.

$$ \sqrt[12]{a^9} \cdot \sqrt[6 \cdot \color{red}2]{(b)^\color{red}2} $$

$$ \sqrt[12]{a^9} \cdot \sqrt[12]{b^2} $$

Now both radicals share the same index.

We can therefore multiply them directly.

$$ \sqrt[12]{a^9} \cdot \sqrt[12]{b^2} = \sqrt[12]{a^9 b^2} $$

The result is

$$ \sqrt[12]{a^9 b^2} $$

Example 2

The following radicals have different indices.

$$ \sqrt[4]{7} \cdot \sqrt[6]{7} \cdot \sqrt[3]{7} $$

Compute the least common multiple (LCM) of the indices.

$$ m.c.m.(4,6,3) = 12 $$

Hence, the common index is 12.

Rewrite each radical using the invariance property of radicals.

$$ \sqrt[4 \cdot 3]{7^3} \cdot \sqrt[6 \cdot 2]{7^2} \cdot \sqrt[3 \cdot 4]{7^4} $$

$$ \sqrt[12]{7^3} \cdot \sqrt[12]{7^2} \cdot \sqrt[12]{7^4} $$

All radicals now share the same index.

Proceed with the multiplication.

$$ \sqrt[12]{7^3 \cdot 7^2 \cdot 7^4} $$

$$ \sqrt[12]{7^{3+2+4}} $$

$$ \sqrt[12]{7^9} $$

Note. This expression already represents the product. However, the radical is reducible because the index and the exponent of the radicand have a common divisor. To complete the simplification, rewrite the result as an equivalent irreducible radical.

Finally, simplify by dividing both the index and the exponent by 3.

$$ \sqrt[\frac{12}{3}]{7^{\frac{9}{3}}} $$

$$ \sqrt[4]{7^3} $$

This is the final result of the multiplication.

And so on.