Question

apply the properties of integer exponents to generate an equivalent algebraic expression to (4r^4y^-4)^3/(z^7y^3)^2

Answer 1

To simplify the expression \((4r^4y^{-4})^3/(z^7y^3)^2\), we will apply the properties of integer exponents step by step.

1. **Expand the Numerator and Denominator**:

First, apply the exponent to both the numerator and denominator:

\[
(4r^4y^{-4})^3 = 4^3(r^4)^3(y^{-4})^3
\]
\[
(z^7y^3)^2 = (z^7)^2(y^3)^2
\]

2. **Calculate Each Part**:

Now, we calculate each of those parts:

- For the numerator:
\[
4^3 = 64
\]
\[
(r^4)^3 = r^{4 \cdot 3} = r^{12}
\]
\[
(y^{-4})^3 = y^{-4 \cdot 3} = y^{-12}
\]

Thus, the numerator simplifies to:
\[
64r^{12}y^{-12}
\]

- For the denominator:
\[
(z^7)^2 = z^{7 \cdot 2} = z^{14}
\]
\[
(y^3)^2 = y^{3 \cdot 2} = y^6
\]

Thus, the denominator simplifies to:
\[
z^{14}y^6
\]

3. **Combine the Results**:

Now, we have the expression:
\[
\frac{64r^{12}y^{-12}}{z^{14}y^6}
\]

4. **Simplify**:

When dividing powers of the same base, we subtract the exponents:
\[
y^{-12 - 6} = y^{-18}
\]

So the expression becomes:
\[
\frac{64r^{12}}{z^{14}y^{18}}
\]

5. **Final Expression**:

Therefore, the simplified equivalent algebraic expression is:
\[
\frac{64r^{12}}{z^{14}y^{18}}
\]

This is the final result for the given expression using the properties of integer exponents.