QuestionWhich of the following is equal to $\left[\frac{\left(x^{2} y^{3}\right)^{-2}}{\left(x^{6} y^{3} z\right)^{2}}\right]^{3}$ ? $\frac{\left(x^{2} y^{3}\right)^{-6}}{\left(x^{6} y^{3} z\right)^{6}}$ $\frac{1}{x^{8} y^{36} z^{6}}$ $\frac{\left(x^{2} y^{3}\right)}{\left(x^{6} y^{3} z\right)^{5}}$ $x^{-12} y^{-18}$

Studdy Solution

STEP 1

1. We are given an expression with exponents and need to simplify it.
2. The expression involves negative exponents and requires careful handling of exponent rules.
3. We need to compare the simplified expression to the given options.

STEP 2

1. Simplify the expression inside the brackets.
2. Apply the outer exponent to the simplified expression.
3. Compare the result with the given options.

STEP 3

Simplify the expression inside the brackets:
The expression is $\left[\frac{\left(x^{2} y^{3}\right)^{-2}}{\left(x^{6} y^{3} z\right)^{2}}\right]$ .
First, simplify $\left(x^{2} y^{3}\right)^{-2}$ :
$\left(x^{2} y^{3}\right)^{-2} = x^{-4} y^{-6}$
Next, simplify $\left(x^{6} y^{3} z\right)^{2}$ :
$\left(x^{6} y^{3} z\right)^{2} = x^{12} y^{6} z^{2}$
Now, substitute these into the expression:
$\frac{x^{-4} y^{-6}}{x^{12} y^{6} z^{2}} = x^{-4 - 12} y^{-6 - 6} z^{-2} = x^{-16} y^{-12} z^{-2}$

STEP 4

Apply the outer exponent of 3 to the simplified expression:
$\left(x^{-16} y^{-12} z^{-2}\right)^{3} = x^{-48} y^{-36} z^{-6}$

STEP 5

Compare the result with the given options:
The simplified expression is $\frac{1}{x^{48} y^{36} z^{6}}$ .
This matches the option:
$\frac{1}{x^{48} y^{36} z^{6}}$
The equivalent expression is:
$\boxed{\frac{1}{x^{48} y^{36} z^{6}}}$

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