Radicals are the opposite of exponents. Instead of multiplying a number by itself multiple times, a radical is a number that must be multiplied by itself several times to get a certain value. For example, $100 $ asks for the number that can be multiplied by itself twice to get $100$, and the answer is $10$ because $10∗10=100$.

There are two rules you should keep in mind when you are working with radicals.

Rule 1: Use multiplication and division to combine or split inside radicals

For example:

$2 ∗3 2 ∗3 2∗32∗3 =6 =2∗3 =2∗3=6 $

The same works for division:

$12 /6 12 /6 12/612/6 =2 =12/6 =12/6=2 $

Rule 2: Treat radicals like variables from the outside

For example:

$32 +22 =52 $

Imagine if $x$ was used to represent the radical $2 $:

$3x+2x32 +22 =5x=52 $

It can be useful to translate radicals into exponents and use exponent rules to solve for the answer. Every radical can be described as a fraction exponent. For example:

$25 =225_{1} =25_{(1/2)}$

The numerator of the exponent is the exponent inside the radical. We can represent 25 with an implied exponent of 1, as anything raised to the power of 1 is itself. All square roots have an implied 2 outside of the radical because a square root asks for the number that, when multiplied by itself, twice gives you the value inside the radical. If there were a 3 outside the radical, it would be a cube root.

Can you rewrite this radical into exponents?

$312_{2} ∗312 $

(spoiler)

First, let’s transform these both into fraction exponents:

$312_{2} ∗312 =12_{2/3}∗12_{1/3}$

When we have fractions in exponents and the denominators are equal, we can simply add them:

$2/3+1/3=1$

Putting it all together:

$12_{2/3}∗12_{1/3}=12_{(2/3)+(1/3)}=12_{1}=12$

All right!

Here’s a more complex question that requires the use of both rules:

Simplify the following equation:

$53 ∗27 −281 −18 $

The trick to solving this question is understanding that we can combine $3 $ and $27 $ into $81 $, and that we can split out $18 $ into its components of $2 $ and $9 $.