Question

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

Answer 1

`8sqrt5`

Explanation:

`8sqrt(5/4)`

`sqrt4= 2` so the fraction looks like `8sqrt5/2`.
The 8 and the 2 can cancel out which leads to `4sqrt5`

`3sqrt20`

`sqrt20 = 2sqrt5`. Then multiply it by 3 which makes `6sqrt5`.

Now the problem looks like this:

`4sqrt5 + 6sqrt5 - 10sqrt(1/5)`

For `10sqrt(1/5)`, we will rationalise the denominator because we cannot have a root as the denominator.

`sqrt(1/5)xxsqrt5 = sqrt5/5`

It is now `10sqrt5/5`

The 10 and the 5 cancel out leaving `2sqrt5`.

The problem is now like this:

`4sqrt5 + 6sqrt5 - 2sqrt5`

`=8sqrt5`

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

Answer 2

`color(red)(=8sqrt5`

Explanation:

`8sqrt(5/4)+3sqrt20-10sqrt(1/5)`

`:.=8 sqrt5/sqrt4+3sqrt(2*2*5)-10sqrt1/sqrt5`

`color(red)(sqrt2*sqrt2=2`

`:.=8sqrt5/sqrt(2*2)+3*2sqrt5-10 1/sqrt5`

`:.=8sqrt5/2+6 sqrt5/1-10/sqrt5`

`:.=(8sqrt5*sqrt5+2sqrt5*6sqrt5-20)/(2sqrt5)`

`:.=(8*5+12*5-20)/(2sqrt5)`

`:.=(40+60-20)/(2sqrt5)`

`:.=80/(2sqrt5)`

`:.=40/sqrt5`

`:.=40/sqrt5 xx sqrt5/sqrt5` rationalise denominator

`:.(cancel40^color(red)8sqrt5)/cancel5^color(red)1`

`:.color(red)(8sqrt5`

Answer 3

Slightly different approach

`8sqrt(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)color(white)(.)(4+6-2)`

`8sqrt(5)`