# How do you simplify 2 sqrt20 + 8 sqrt45 - sqrt80?

Apr 15, 2018

The answer is $24 \sqrt{5}$.

#### Explanation:

Note: when the variables a, b, and c are used, I am referring to a general rule that will work for every real value of a, b, or c.

You can use the rule $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ to your advantage:

$2 \sqrt{20}$ equals $2 \sqrt{4 \cdot 5}$, or $2 \sqrt{4} \cdot \sqrt{5}$.

Since $\sqrt{4} = 2$, you can substitute $2$ in to get $2 \cdot 2 \cdot \sqrt{5}$, or $4 \sqrt{5}$.

Use the same rule for $8 \sqrt{45}$ and $\sqrt{80}$:

$8 \sqrt{45} \to 8 \sqrt{9 \cdot 5} \to 8 \sqrt{9} \cdot \sqrt{5} \to 8 \cdot 3 \cdot \sqrt{5} \to 24 \sqrt{5}$.

$\sqrt{80} \to \sqrt{16 \cdot 5} \to \sqrt{16} \cdot \sqrt{5} \to 4 \sqrt{5}$.

Substitute these into the original equation and you get:

$4 \sqrt{5} + 24 \sqrt{5} - 4 \sqrt{5}$.

Since $a \sqrt{c} + b \sqrt{c} = \left(a + b\right) \sqrt{c}$, and likewise $a \sqrt{c} - b \sqrt{c} = \left(a - b\right) \sqrt{c}$, you can simplify the equation:

$4 \sqrt{5} + 24 \sqrt{5} - 4 \sqrt{5} \to 28 \sqrt{5} - 4 \sqrt{5} \to 24 \sqrt{5}$ , the final answer.

Hope this helps!