Question

select the correct answer.
which expression simplifies to $5\sqrt{3}$?
a. $\sqrt{30}$
b. $\sqrt{45}$
c. $\sqrt{75}$
d. $\sqrt{15}$

Explanation:

Step1: Recall the property of square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ (where $a \geq 0$, $b \geq 0$)

We need to simplify each option to see which one equals $5\sqrt{3}$.

Step2: Simplify Option A: $\sqrt{30}$

Factor 30: $30 = 2 \times 3 \times 5$. There are no perfect square factors, so $\sqrt{30}$ cannot be simplified to $5\sqrt{3}$.

Step3: Simplify Option B: $\sqrt{45}$

Factor 45: $45 = 9 \times 5$. Then $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$. This is not equal to $5\sqrt{3}$.

Step4: Simplify Option C: $\sqrt{75}$

Factor 75: $75 = 25 \times 3$. Then $\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$. This matches the desired expression.

Step5: Simplify Option D: $\sqrt{15}$

Factor 15: $15 = 3 \times 5$. There are no perfect square factors, so $\sqrt{15}$ cannot be simplified to $5\sqrt{3}$.

Answer:

C. $\sqrt{75}$