Question

How do you simplify `( sqrt32)/ (5 sqrt 14)`?

Answer 1

`(4sqrt(7))/35`

Explanation:

You have to try and spot common values that can be cancelled out.

Both 32 and 14 are even so 2 has to be a common factor giving:

`" "(sqrt(2xx16))/(5sqrt(2xx7))`

7 is a prime number so the denominator can not be broken down any further

However, 16 in the numerator can be broken down into `4^2` so we now have:

`" "(sqrt(2xx4^2))/(5sqrt(2xx7))`

`" "1/5xxsqrt(2)/(sqrt(2)) xxsqrt(4^2)/sqrt(7)`

`" " 1/5xx 1 xx 4/sqrt(7)" "=" "4/(5sqrt(7))`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Added extension to solution
I forgot that standard practice is that you try to remove any roots from the denominator!

Multiply by 1 but in the form of `sqrt(7)/sqrt(7)` giving:

`" "4/(5sqrt(7))xx sqrt(7)/sqrt(7)" "=" "(4sqrt(7))/(5xx7)" "=" "(4sqrt(7))/35`

Answer 2

`sqrt 32/(5sqrt 14)=(4sqrt7)/35`

Explanation:

`sqrt 32/(5sqrt 14)`

Simplify `sqrt 32`.

`sqrt(2xx2xx2xx2xx2)=`

`sqrt(2^2xx2^2xx2)=`

`4sqrt2`

Add this back into the expression.

`(4sqrt 2)/(5sqrt14)`

Rationalize the denominator by multiplying both the numerator and denominator by `sqrt 14`.

`(4sqrt 2)/(5sqrt14)xx(sqrt14)/sqrt 14=`

`(4sqrt 2sqrt14)/(5xx14)`

Simplify the numerator by multiplying the square roots.

`(4sqrt 28)/(5xx14)`

Simplify the square root by factoring.

`(4sqrt 28)/(5xx14)=(4sqrt(2xx2xx7))/(5xx14)=(4xx2sqrt7)/(5xx14)=(8sqrt7)/(5xx14)`

Simplify the denominator.

`(8sqrt 7)/(70)`

Simplify the expression.

`(4sqrt7)/35`