Question

How do you simplify `sqrt[81x^12y^8z^10]`?

Answer 1

`sqrt(81x^12y^8z^10)= 9x^6y^4z^5`

Explanation:

You can use the following rule:

`sqrt(ab) = sqrt(a) xx sqrt(b)`

In order to see how to simplify `sqrt(81x^12y^8z^10)`, we can split it as follows:

`sqrt(81x^12y^8z^10)`
`=sqrt(81) xx sqrt(x^12) xx sqrt(y^8) xx sqrt(z^10)`

Remember that `sqrt(a) = a^(1/2)`, so:
`sqrt(a^n)`
`=a^(n^(1/2))`
`=a^(n/2)`

In other words, square rooting an expression with an exponent halves the exponent.

`sqrt(81) xx sqrt(x^12) xx sqrt(y^8) xx sqrt(z^10)`
`= 9 xx x^6 xx y^4 xx z^5`

`= 9x^6y^4z^5`

Note: usually teachers will be fine with you skipping straight from `sqrt(81x^12y^8z^10)` to `9x^6y^4z^5`. You don't need to show the long process all the time unless your teacher asks for it.

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Answer 2

`sqrt[81x^12y^8z^10] = color(green)(9*x^(6)*y^(4)*z^(5)`

Explanation:

There are two simple Exponents rules we need to know to answer this question

1) `color(blue)(sqrt(a*b*c) = sqrta*sqrtb*sqrtc`

2) `color(blue)(sqrt(a^m) = a^(m/2)`

Based on the first rule,

`sqrt[81x^12y^8z^10] = sqrt 81*sqrt(x^12)*sqrt(y^8)*sqrt(z^10)`

`= sqrt(9^2)*sqrt(x^12)*sqrt(y^8)*sqrt(z^10)`

Based on the second rule, we write the above expression as

`= 9^(2/2)*x^(12/2)*y^(8/2)*z^(10/2)`

`= color(green)(9*x^(6)*y^(4)*z^(5)`