Question

How do you simplify `\frac { ( x ^ { 2} y ^ { 5} z ^ { 6} ) ^ { 7} } { ( x ^ { 6} y ^ { 5} z ) ^ { 6} }`?

Answer 1

`(y^5z^36)/x^22`

Explanation:

1) Distribute the exponents
`(x^14y^35z^42)/(x^36y^30z^6)`

2) Simplify
`(y^5z^36)/x^22`

Answer 2

`x^(−22) y^5 z^36`

Explanation:

Simplify

`((x^2 y^5 z^6)^7) / ((x^6 y^5 z)^6)`

First raise all the factors to the powers outside the parentheses.
Then cancel factors that appear in the numerator and the denominator.

Raise the expressions inside the parentheses to the powers outside.

To raise a power to a power, you multiply

1) Starting with the numerator, raise the exponents inside the parentheses to the 7th power by multiplying each exponent by `7`

After you have multiplied the exponents of the expression in the numerator, you will have this:

`(x^14 y^35 z^42) / ((x^6  y^5 z^1)^6)`

2) Now working with the denominator, raise each of the powers to the 6th power by multiplying each exponent by 6.

After you have multiplied, you will have this:

`(x^14 y^35 z^42) / (x^36 y^30 z^6)`

Now cancel all the factors that appear in both the numerator and in the denominator

To divide exponents, you subtract

`x^(14-36)  y^(35-30)  z^(42-6)`

After you subtract, you will get this:

`x^-22 y^5  z^ 36` `larr` answer

`color(white)( . . . . . . . . . . .)`. . . . . . . . . .

Note:

If you don't want a negative exponent in the answer, you can change
the negative sign to a positive by writing `x^(+22)` in the denominator, like this:

`(y^5 z^ 36) / (x^22)` `larr` same answer