simplify : `\left(4x^{3}y^{-4}\right)^{-2}`.

Answer 1

To simplify the expression $(4x^3{y}^{-4}{)}^{-2}$, we will apply the property of exponents $(a^m{)}^{-n}=a^{-mn}$.

Starting with the expression:

$$(4x^3{y}^{-4}{)}^{-2}$$

we can distribute the exponent $-2$ to each term inside the parentheses:

$$=4^{-2}(x^3{)}^{-2}(y^{-4}{)}^{-2}$$

Now, let's simplify each part:

1. $4^{-2}=\frac{1}{4^2}=\frac{1}{16}$
2. $(x^3{)}^{-2}=x^{-6}$
3. $(y^{-4}{)}^{-2}=y^8$

Putting this all together, we get:

$$=\frac{1}{16}\cdot x^{-6}\cdot y^8$$

Now, we can express $x^{-6}$ as $\frac{1}{x^6}$:

$$=\frac{y^8}{16x^6}$$

Thus, the simplified form of the expression $(4x^3{y}^{-4}{)}^{-2}$ is:

$$\frac{y^8}{16x^6}$$