QuestionSimplify this expression: $\left(4 x^{3} y^{-4}\right)^{-2}$ .

Studdy Solution

STEP 1

1. We are asked to simplify the expression.
2. The expression involves negative exponents and a power raised to another power.

STEP 2

1. Apply the power of a power rule to simplify the expression.
2. Simplify the expression by applying the negative exponent rule.

STEP 3

Apply the power of a power rule, which states $(a^m)^n = a^{m \cdot n}$ , to each component inside the parentheses:
$\left(4 x^{3} y^{-4}\right)^{-2} = 4^{-2} \cdot (x^3)^{-2} \cdot (y^{-4})^{-2}$
This results in:
$4^{-2} \cdot x^{-6} \cdot y^{8}$

STEP 4

Simplify each term using the negative exponent rule, which states $a^{-n} = \frac{1}{a^n}$ :
$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$
$x^{-6} = \frac{1}{x^6}$
$y^{8} \text{ remains as } y^{8}$
Combine these results:
$\frac{1}{16} \cdot \frac{1}{x^6} \cdot y^8 = \frac{y^8}{16x^6}$
The simplified expression is:
$\boxed{\frac{y^8}{16x^6}}$

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