Question

How do you rationalize the denominator and simplify `(5sqrt6)/sqrt10`?

Answer 1

`sqrt15`

Explanation:

multiply by `sqrt10`:

`(5sqrt6sqrt10)/((sqrt10)^2)`

`sqrt6 * sqrt10 = sqrt (6*10) = sqrt60`

`(sqrt10)^2 = 10`

`(5sqrt6)/(sqrt10) = (5sqrt60)/10`

`sqrt60 = sqrt4 * sqrt15 = 2sqrt15`

`5sqrt60 = 5 * 2 * sqrt15 = 10sqrt15`

`(5sqrt60)/10 = (10sqrt15)/10`

`= (sqrt15)/1`

`= sqrt15`

Answer 2

`sqrt15`

Explanation:

In order to rationalize the denominator, you can multiply by `sqrt10/sqrt10`. This is the same as multiplying the fraction by one. If you multiply `(5sqrt6)/sqrt10 *sqrt10/sqrt10`, you get `(5sqrt60)/10`. If you multiply the square roots in the denominator, you get `sqrt100`, which is equivalent to 10.

With `(5sqrt60)/10`, you can simplify to just `sqrt60/2`.

Next, you can simplify `sqrt60` by doing a factor tree. When you do a factor tree, you will find you can pull out a factor of 2 from the square root leaving you with `(2sqrt15)/2`.

Lastly, just cancel out the 2 in the numerator and denominator, and you get the answer of `sqrt15`

Answer 3

`sqrt15`

Explanation:

`"using the "color(blue)"laws of radicals"`

`•color(white)(x)sqrtaxxsqrtbhArrsqrtab`

`•color(white)(x)sqrtaxxsqrta=a`

`"To rationalise the denominator that is eliminate the "`
`"radical from the denominator"`

`"multiply numerator/denominator by "sqrt10`

`rArr(5sqrt6)/sqrt10`

`=(5xxsqrt6xxsqrt10)/(sqrt10xxsqrt10)`

`=(5xxsqrt60)/10`

`=(5xxsqrt(4xx15))/10`

`=(5xxsqrt4xxsqrt15)/10`

`=(5xx2xxsqrt15)/10=(cancel(10)sqrt15)/cancel(10)=sqrt15`