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

1 answer

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:

\[ \frac{y^{8}}{16x^{6}} \]