Question

How do you simplify `(9a^3b^4)^(1/2)`?

Answer 1

`3b^2a^(3/2)` or `3ab^2sqrta` depending on what is meant by "simplify".

Explanation:

Exponents distribute across multiplication, so

`(9a^3b^4)^(1/2) = 9^(1/2)(a^3)^(1/2)(b^4)^(1/2)`

Now use `9^(1/2) = sqrt9 = 3`.

Also use `(x^a)^b = x^(ab)`, to get

`(a^3)^(1/2) = a^(3/2)` which can be written `a*a^(1/2) = a sqrta`

and `(b^4)^(1/2) = b^2`.

`(9a^3b^4)^(1/2) = 9^(1/2)(a^3)^(1/2)(b^4)^(1/2)`

`= 3a^(3/2)b^2` `" "` (which is simpler in some sense)

`= 3asqrtab^2`

which is not as easy to read as putting the radical last

`= 3ab^2sqrta`.

Answer 2

`3a^(3/2)b^2`

Explanation:

Using the `color(blue)"laws of exponents"`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)((a^m)^n=a^(mn))color(white)(a/a)|)))`

This law applies to each value inside the bracket.

`rArr9^(1xx1/2xxa^(3xx1/2)xxb^(4xx1/2)=9^(1/2)a^(3/2)b^2`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(a^(1/2)=sqrta)color(white)(a/a)|)))`

`rArr9^(1/2)=sqrt9=3`

`rArr(9a^3b^4)^(1/2)=3a^(3/2)b^2`