#### beachgirl

##### New member
Simplify the exponents

(3X^2Y^-2)^-3/(3X^-1Y)^-2

#### tkhunny

##### Moderator
Staff member
Use this rule, first: $$\displaystyle x^{-a} = \frac{1}{x^{a}}$$

Show us what you get.

#### beachgirl

##### New member
(3)^-3X^-6Y^6/(3)^-2X^2Y^-2

I don't know what to do past this step. And I am not even sure if this is correct.

Thanks

#### tkhunny

##### Moderator
Staff member
You didn't use the rule I showed you. You used this one:

$$\displaystyle (x^{a})^{b} = x^{a*b}$$

Try this more complicated version of the first rule:

$$\displaystyle \frac{x^{-a}}{y^{-b}} = \frac{y^{b}}{x^{a}}$$

#### mmm4444bot

##### Super Moderator
Staff member
beachgirl said:
(3)^-3X^-6Y^6/(3)^-2X^2Y^-2

… I am not even sure if this is correct.

[3^(-3) X^(-6) Y^6] / [3^(-2) X^2 Y^(-2)]

Yes, it is.

The notation is better when you put the parentheses around negative exponents, instead of putting them around the base.

The properties of exponents used to simplify expressions like this exercise can be carried out in different orders, so there's more than one way to simplify.

From here, you could either proceed using the rule(s) already provided by TK, or you could use another property.

a^n/a^m = a^(n - m)

In other words, when the base is the same on top and bottom, you may eliminate the fraction by subtracting the bottom exponent from the top exponent.

If you still have negative exponents, then bring back the fraction as the final step.

EG:

[4^5 a^(-5) b^6] / [4^(-1) a^(-5) b^8]

4^(5 + 1) a^(-5 + 5) b^(6 - 8)

4^6 a^0 b^(-2)

4096/b^2