Question

(a) Simplify $\sqrt{\frac{27}{3}}$and explain all your steps.

(b) Simplify $\sqrt{\frac{27}{5}}$and explain all your steps.

(c) Why are the two methods of simplifying square roots different?

Short answer

(a) The simplified form of $\sqrt{\frac{27}{3}}$is $3$.

(b) The simplified form of $\sqrt{\frac{27}{5}}$is $\frac{\sqrt{135}}{5}$

(c) The two methods of simplifying square roots are different as $\sqrt{\frac{27}{3}}$is a perfect square and$\sqrt{\frac{27}{5}}$is not a perfect square.

Step-by-step solution

Step 1. Part (a) Simplify 273 and explain all the steps.

Step 1- Write expression.

$\sqrt{\frac{27}{3}}$

Step 2- Simplify the expression under square root.

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

Step 2. Part (b) Simplify 275 and explain all the steps.

Step 1- Write expression.

$\sqrt{\frac{27}{5}}$

Step 2- Rewrite using quotient property.

$\frac{\sqrt{27}}{\sqrt{5}}$

Step 3- Rationalize the denominator by multiplying numerator and denominator by $\sqrt{5}$.

$$\begin{gathered} \frac{\sqrt{27}}{\sqrt{5}}\times \frac{\sqrt{5}}{\sqrt{5}} \\ =\frac{\sqrt{135}}{5} \end{gathered}$$

Step 3. Part (c) Why are the two methods of simplifying square roots different?

The two methods of simplifying square roots are different as $\sqrt{\frac{27}{3}}$ is a perfect square and $\sqrt{\frac{27}{5}}$ is not a perfect square.