Square roots and radicals

Radicals are the opposite of exponents. With exponents, you multiply a number by itself multiple times. With radicals, you’re looking for the number that must be multiplied by itself a certain number of times to produce a given value.

For example, $\sqrt{100}$ asks for the number that can be multiplied by itself twice to get $100$. The answer is $10$ because $10\ast 10=100$.

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

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

For example:

$$\begin{array}{rl}\sqrt{2}\ast \sqrt{3}& =\sqrt{6}\\ \sqrt{2}\ast \sqrt{3}& =\sqrt{2\ast 3}\\ 2\ast 3& =2\ast 3\\ 2\ast 3& =6\end{array}$$

The same works for division:

$$\begin{array}{rl}\sqrt{12}/\sqrt{6}& =\sqrt{2}\\ \sqrt{12}/\sqrt{6}& =\sqrt{12/6}\\ 12/6& =12/6\\ 12/6& =2\end{array}$$

Rule 2: Treat radicals like variables from the outside

For example:

$$\begin{array}{rl}3\sqrt{2}+2\sqrt{2}& =5\sqrt{2}\end{array}$$

This works because you can think of the radical $\sqrt{2}$ as a single “thing,” like a variable. If $x$ represents $\sqrt{2}$, then:

$$\begin{array}{rl}3x+2x& =5x\\ 3\sqrt{2}+2\sqrt{2}& =5\sqrt{2}\end{array}$$

It can also be useful to rewrite radicals as exponents and then use exponent rules. Every radical can be written as a fractional exponent. For example:

$$\sqrt{25}=\sqrt[2]{25^1}=25^{(1/2)}$$

Here’s how to interpret that exponent:

• The numerator is the exponent inside the radical. In this example, $25$ has an implied exponent of $1$ because anything raised to the power of $1$ is itself.
• The denominator is the index of the radical. A square root has an implied index of $2$ because you multiply the answer by itself twice to get the value inside the radical. If the index were $3$, it would be a cube root.

Can you rewrite this radical expression using exponents?

$$\sqrt[3]{12^2}\ast \sqrt[3]{12}$$

All right!

Here’s a more complex question that uses both rules:

Simplify the following equation:

$5\sqrt{3}\ast \sqrt{27}-2\sqrt{81}-\sqrt{18}$

The key ideas are:

• You can combine $\sqrt{3}$ and $\sqrt{27}$ into a single radical.
• You can split $\sqrt{18}$ into factors that include a perfect square.

$$\begin{gathered} 5\sqrt{3}\ast \sqrt{27}-2\sqrt{81}-\sqrt{18} \\ 5\sqrt{3\ast 27}-2\sqrt{81}-\sqrt{2\ast 9} \\ 5\sqrt{81}-2\sqrt{81}-\sqrt{2}\ast \sqrt{9} \\ 5\ast 9-2\ast 9-\sqrt{2}\ast 3 \\ 45-18-3\sqrt{2} \\ 27-3\sqrt{2} \end{gathered}$$

So, the equation simplifies to $27-3\sqrt{2}$.

Bringing it all together: question walkthrough video

Here’s a video going through one of our practice questions to show these ideas in action: