Question

How do you simplify `\frac { \sqrt { 12} \cdot \sqrt { 3} } { 2} + \frac { \sqrt { 128} } { \sqrt { 2} }`?

Answer 1

See a solution process below:

Explanation:

First, in order to add the fractions, we need to put the fractions over a common denominator by multiplying the fraction on the right by the appropriate form of `1`:

`(sqrt(12) * sqrt(3))/2 + (sqrt(2)/sqrt(2) * sqrt(128)/sqrt(2)) =>`

`(sqrt(12) * sqrt(3))/2 + (sqrt(2) * sqrt(128))/(sqrt(2) * sqrt(2)) =>`

`(sqrt(12) * sqrt(3))/2 + (sqrt(2) * sqrt(128))/2`

Next we can use these rules for radicals to simplify the numerators of each fraction:

`sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))` and `sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))`

`(sqrt(color(red)(12)) * sqrt(color(blue)(3)))/2 + (sqrt(color(red)(2)) * sqrt(color(blue)(128)))/2 =>`

`sqrt(color(red)(12) * color(blue)(3))/2 + sqrt(color(red)(2) * color(blue)(128))/2 =>`

`sqrt(36)/2 + sqrt(256)/2 =>`

`6/2 + 16/2 =>`

`3 + 8 =>`

`11`

Answer 2

This simplifies all the way down to 11.
Details below.

Explanation:

Treat these terms as you would any other fraction - first you create a common denominator, then add the numerators.
To do that, you can multiply numerator and denominator by the same factor, without changing the value of the fraction.
Specifically, multiply the second term by `sqrt2` in the top and bottom.
Now, your terms look like this:
`(sqrt12*sqrt3)/2+(sqrt2*sqrt128)/2`

Now, use the fact that `sqrta*sqrtb=sqrt(ab)` to write `sqrt2*sqrt(128)` as `sqrt(256)`, which is 16.

Likewise, `(sqrt12*sqrt3)= sqrt36` which is 6
The sum is now `6/2 + 16/2`

or `3 + 8=11`