Question

How do you simplify `(root3(6)*root4(6))^12`?

Answer 1

`(root3(6)*root(4)6)^12=6^7=279936`

Explanation:

We need the rules:

• `rootmx=x^(1/m)" "" "color(red)star`
• `x^a*x^b=x^(a+b)" "" "color(green)star`
• `(x^c)^d=x^(cd)" "" "color(blue)star`

Using `color(red)star`, we see that:

`(root3(6)*root(4)6)^12=(6^(1/3)*6^(1/4))^12`

Now using `color(green)star`, this becomes:

`(6^(1/3)*6^(1/4))^12=(6^(1/3+1/4))^12`

Note that `1/3+1/4=4/12+3/12=7/12`.

`(6^(1/3+1/4))^12=(6^(7/12))^12`

Now using `color(blue)star`, we multiply the exponents:

`(6^(7/12))^12=6^(7/12xx12)`

And we see that `7/12xx12=7`:

`6^(7/12xx12)=6^7`

All of which we did without a calculator! For an expanded value, we could plug in `6^7` into a calculator to see that `6^7=279936`.