# How do you simplify 3sqrt5 (times) sqrt5 + 3sqrt5 (times) 2sqrt75?

Sep 30, 2017

Depending on how the question is interpreted you have:

$45 + \sqrt{15} \text{ or } 120 \sqrt{5}$

#### Explanation:

$\textcolor{b l u e}{\text{Assumption: you really meant the question to be as written}}$

color(brown)("3sqrt(5)xxsqrt(5)+3sqrt(5)xx2sqrt(75)

As you there is no grouping by brackets we have to look at priority of action (add, divide, multiply etc). Multiplication has a higher priority than add so we have to apply that first. Thus we have:

$\left(3 \sqrt{5} \times \sqrt{5}\right) + \left(3 \sqrt{5} \times 2 \sqrt{75}\right)$

$\left(3 \times {\left(\sqrt{5}\right)}^{2}\right) + \left(3 \times 2 \times \sqrt{5} \times \sqrt{75}\right)$

note that $75 \to 5 \times 15$ so $\sqrt{5} \times \sqrt{75} \to {\left(\sqrt{5}\right)}^{2} \times \sqrt{15}$

$\left(3 \times {\left(\sqrt{5}\right)}^{2}\right) + \left(3 \times 2 \times {\left(\sqrt{5}\right)}^{2} \times \sqrt{15}\right)$

$\textcolor{w h i t e}{\text{dddd")(15)color(white)("ddd")+color(white)("ddd}} \left(30 + \sqrt{15}\right)$

$45 + \sqrt{15}$

Note that $15 = 3 \times 5$ and both 3 and 5 prime numbers so it is simpler to leave the $\sqrt{15}$ as it is.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Assumption: the brackets are the wrong way round}}$
Having the brackets round the way you have is very unexpected.

$\textcolor{b r o w n}{\left(3 \sqrt{5}\right) \times \left(\sqrt{5} + 3 \sqrt{5}\right) \times \left(2 \sqrt{5}\right)}$

$\left(3 \sqrt{5}\right) \times \textcolor{w h i t e}{\text{dd")(4sqrt(5))color(white)("ddd}} \times \left(2 \sqrt{5}\right)$

Dealing with the whole numbers part $\to 3 \times 4 \times 2 = \textcolor{red}{24}$

Dealing with the square roots part $\textcolor{w h i t e}{\text{d}} \to \left(\sqrt{5} \times \sqrt{5}\right) \times \sqrt{5}$

$\textcolor{w h i t e}{\text{dddddddddddddddddddddddddd")->color(white)("dddd") 5color(white)("dddd}} \times \sqrt{5} = \textcolor{p u r p \le}{5 \sqrt{5}}$

Putting it all together we have: $\textcolor{red}{24} \textcolor{p u r p \le}{\times 5 \sqrt{5}} = 120 \sqrt{5}$