Question

How do you multiply `sqrt(2)(sqrt 8 + sqrt 4)`?

Answer 1

`sqrt(2)(sqrt(8)+sqrt(4)) = 4+2sqrt(2)`

Explanation:

Method 1: Multiply first then simplify
The distributive property (of multiplication over addition) tells us that
`color(white)("XXXX")` `a*(b+c) = ab+ac`

Further as a property of exponents (and therefore of square roots)
`color(white)("XXXX")` `a^m*b^m = (a*b)^m`
`color(white)("XXXX")` `sqrt(a)*sqrt(b) = sqrt(a*b)` since `(sqrt(k) = k^(1/2))`

So
`sqrt(2)(sqrt(8)+sqrt(4))`
`color(white)("XXXX")` `=sqrt(2)*sqrt(8) + sqrt(2)*sqrt(4)`

`color(white)("XXXX")` `=sqrt(2*8) + sqrt(2*4)`

`color(white)("XXXX")` `=sqrt(16) + sqrt(8)`

`color(white)("XXXX")` `=4+ 2sqrt(2)`

Method 2: Simplify roots then multiply
`sqrt(2)(sqrt(8)+sqrt(4))`
`color(white)("XXXX")` `=sqrt(2)(2sqrt(2)+2)`

`color(white)("XXXX")` `=2sqrt(2)*sqrt(2) + 2sqrt(2)`

`color(white)("XXXX")` `=2*2 + 2sqrt(2)`

`color(white)("XXXX")` `=4+2sqrt(2)`

Answer 2

`sqrt(2)(sqrt 8 + sqrt 4)=2(2+sqrt(2))`

Explanation:

1. `sqrt(2)(sqrt 8 + sqrt 4)=sqrt(2)*sqrt(8)+sqrt(2)*sqrt(4)` for the distributive property.
2. `sqrt(2)*sqrt(8)+sqrt(2)*sqrt(4)=sqrt(2*8)+sqrt(2*4)=sqrt(2^4)+sqrt(2^3)` because we can write the product of two square roots as the square roots of the product. I rewrote the products in exponential form, it is easier.
3. Now, if you remember, the square root of x means x at the exponent of 1/2: `sqrt(x)=x^(1/2)`.
   So in this case we can write:
   `sqrt(2^4)+sqrt(2^3)=2^(4/2)+2^(3/2)`
4. Now we can solve the equation:
   `2^(4/2)+2^(3/2)=2^2+2^(1+1/2)=4+2*2^(1/2)=4+2sqrt(2)=2(2+sqrt(2))`.