Q. 8.26 - Intermediate Algebra

Question

Simplify $\sqrt{45}$

Step-by-Step Solution

Answer

The value of the expression is $3\sqrt{5}$.

Step 1. Given and explanation.

We have $\sqrt{45}$.

Firstly, we will find the largest perfect square factor of the given number in the radicand.

Then using it, we will rewrite the expression as the product of two factors.

Then using the product rule, we will rewrite the radical as the product if two radicals. The simplifying the expression will give us the value.

Step 2. Rewriting radical as product of two radicals using a perfect square.

We have $\sqrt{45}$.

We have $9$ as the largest perfect square factor of $45$ as $9\times 5=45$ and
${\left(3\right)}^2=9$.

So we will rewrite the radical as product of two radicals using the product rule.

$$\begin{gathered} \sqrt{45}=\sqrt{9\times 5} \\ =\sqrt{9}\times \sqrt{5} \end{gathered}$$

Step 3. Simplifying the expression.

We have $\sqrt{9}\times \sqrt{5}$.

We can write $\sqrt{9}$ as $3$ because $\sqrt{9}=\sqrt{3\times 3}$

So using it, we get the value as

$$\begin{gathered} =\sqrt{9}\times \sqrt{5} \\ =3\sqrt{5} \end{gathered}$$

Thus the value of the expression is $3\sqrt{5}$.