Reduced Radical Form

Radicals can be a little daunting, but there are a few easy tricks that can make problems involving radicals much easier to solve. One trick is to change radicals into their reduced radical form. This can be useful in a range of different situations. But how does it work? Let's find out:

What is reduced radical form?

If we see an expression with a radical sign $\left(\sqrt{}\right)$ , we know it is in reduced radical form if the number underneath the radical sign does not contain any perfect squares as factors. Remember that we call the number under the radical sign the "radicand." The same applies to cubic roots, but the radicand in this case cannot contain any perfect cubes as factors.

Remember that perfect squares are numbers that are the squares of other numbers. Examples include $2^2=4$ , $3^2=9$ , etc. The same general logic applies to perfect cubes, such as $3^3=27$ .

Changing radicals into reduced radical form can help us simplify square roots, and it may be very useful in a range of different situations.

The product property of square roots

But how do we find out whether a radicand "contains" any perfect squares?

Easy: We use the product property of square roots. This property states that:

For all real numbers a and b, $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$ .

Reduced radical form: practice questions

a. What if we wanted to simplify $\sqrt{18}$ ?

First, break the radicand into perfect squares and other factors:

$\sqrt{18}=\sqrt{9\times 2}or\sqrt{9}\times \sqrt{2}$

We can simplify this as $3\sqrt{2}$ .

b. What if we wanted to simplify $\sqrt{252}$ ?

Break the radicand into perfect squares and other factors:

$\sqrt{252}=\sqrt{36\times 7}$ or $\sqrt{36}\times \sqrt{7}$

We can simplify this as:

$6\sqrt{7}$

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