Question

1. when $5\sqrt{20}$ is written in simplest form, the result is $k\sqrt{5}$. what is the value of $k$?

a. 20
b.10
c.7
d.4

Explanation:

Step1: Simplify the square root

We know that \( \sqrt{20} = \sqrt{4\times5} \), and by the property of square roots \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (where \( a = 4 \) and \( b = 5 \)), we get \( \sqrt{4\times5}=\sqrt{4}\times\sqrt{5} \). Since \( \sqrt{4} = 2 \), then \( \sqrt{20}=2\sqrt{5} \).

Step2: Substitute back into the original expression

The original expression is \( 5\sqrt{20} \). Substituting \( \sqrt{20}=2\sqrt{5} \) into it, we get \( 5\times(2\sqrt{5}) \).

Step3: Simplify the expression

Multiplying 5 and 2, we have \( 5\times2\sqrt{5}=10\sqrt{5} \). Comparing this with \( k\sqrt{5} \), we can see that \( k = 10 \).

Answer:

b. 10