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.