How do you simplify sqrt125 + sqrt80?

May 28, 2016

The result is $9 \sqrt{5}$.

Explanation:

When you want to simplify the roots, the best thing you can do is to write the factors of your numbers. In your case $125 = 5 \cdot 5 \cdot 5$ and $80 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5$.
The square root is an operation that if you have the same numbers, gives you back one of the two numbers.
As an example $\sqrt{4 \cdot 4} = 4$.
So we apply this property of the square root grouping the numbers in pairs (when possible) and removing from the square root:

$\sqrt{125} = \sqrt{5 \cdot 5 \cdot 5} = \sqrt{\left(5 \cdot 5\right) \cdot 5} = 5 \sqrt{5}$
Here I grouped a pair of $5$ and I applied the rule of the square root. Unfortunately the last $5$ is alone and I have to live it under the root. Now for the second term

$\sqrt{80} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5} = \sqrt{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) \cdot 5} = 2 \cdot 2 \cdot \sqrt{5} = 4 \sqrt{5}$

Again I extracted the pairs from the root.

Finally I have: $\sqrt{125} + \sqrt{80} = 5 \sqrt{5} + 4 \sqrt{5} = 9 \sqrt{5}$.