# How do you simplify the expression 8sqrt(5/4)+3sqrt20-10sqrt(1/5)?

Apr 7, 2017

$8 \sqrt{5}$

#### Explanation:

$8 \sqrt{\frac{5}{4}}$

$\sqrt{4} = 2$ so the fraction looks like $8 \frac{\sqrt{5}}{2}$.
The 8 and the 2 can cancel out which leads to $4 \sqrt{5}$

$3 \sqrt{20}$

$\sqrt{20} = 2 \sqrt{5}$. Then multiply it by 3 which makes $6 \sqrt{5}$.

Now the problem looks like this:

$4 \sqrt{5} + 6 \sqrt{5} - 10 \sqrt{\frac{1}{5}}$

For $10 \sqrt{\frac{1}{5}}$, we will rationalise the denominator because we cannot have a root as the denominator.

$\sqrt{\frac{1}{5}} \times \sqrt{5} = \frac{\sqrt{5}}{5}$

It is now $10 \frac{\sqrt{5}}{5}$

The 10 and the 5 cancel out leaving $2 \sqrt{5}$.

The problem is now like this:

$4 \sqrt{5} + 6 \sqrt{5} - 2 \sqrt{5}$

$= 8 \sqrt{5}$

Tip: Dealing with these types of problems, break them into small chunks to make your life easier.

Apr 7, 2017

color(red)(=8sqrt5

#### Explanation:

$8 \sqrt{\frac{5}{4}} + 3 \sqrt{20} - 10 \sqrt{\frac{1}{5}}$

$\therefore = 8 \frac{\sqrt{5}}{\sqrt{4}} + 3 \sqrt{2 \cdot 2 \cdot 5} - 10 \frac{\sqrt{1}}{\sqrt{5}}$

color(red)(sqrt2*sqrt2=2

$\therefore = 8 \frac{\sqrt{5}}{\sqrt{2 \cdot 2}} + 3 \cdot 2 \sqrt{5} - 10 \frac{1}{\sqrt{5}}$

$\therefore = 8 \frac{\sqrt{5}}{2} + 6 \frac{\sqrt{5}}{1} - \frac{10}{\sqrt{5}}$

$\therefore = \frac{8 \sqrt{5} \cdot \sqrt{5} + 2 \sqrt{5} \cdot 6 \sqrt{5} - 20}{2 \sqrt{5}}$

$\therefore = \frac{8 \cdot 5 + 12 \cdot 5 - 20}{2 \sqrt{5}}$

$\therefore = \frac{40 + 60 - 20}{2 \sqrt{5}}$

$\therefore = \frac{80}{2 \sqrt{5}}$

$\therefore = \frac{40}{\sqrt{5}}$

$\therefore = \frac{40}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$ rationalise denominator

$\therefore \frac{{\cancel{40}}^{\textcolor{red}{8}} \sqrt{5}}{\cancel{5}} ^ \textcolor{red}{1}$

:.color(red)(8sqrt5

Apr 9, 2017

Slightly different approach

$8 \sqrt{5}$

#### Explanation:

Multiply by 1 and you do not change the value. But 1 comes in many forms. So you can change the way something looks without changing its intrinsic value.

Build a common factor by trying to have a $\sqrt{5}$ in all the numerators:

color(green)([8sqrt(5/4)color(white)(.) ] + [3sqrt(20)] -[10sqrt(1/5)color(red)(xx1)]

color(green)([8sqrt(5/4)color(white)(.) ] + [3sqrt(20)] -[10sqrt(1/5)color(red)(xxsqrt(5)/sqrt(5)color(white)(.))]

color(green)([8(sqrt(5))/(sqrt(4))color(white)(.) ] + [3sqrt(4xx5)] -[10(sqrt(1))/(sqrt(5))color(red)(xxsqrt(5)/sqrt(5)color(white)(.))]

color(green)([4sqrt(5)color(white)(.)]color(white)(.) +color(white)(..) [6sqrt(5)color(white)(.)]color(white)(.)-" "[2sqrt(5)color(white)(.)]

$\sqrt{5} \textcolor{w h i t e}{.} \left(4 + 6 - 2\right)$

$8 \sqrt{5}$