Apply the properties of integer exponents to generate an equivalent, algebraic expression to (4x power of four y -power of 4 ) and then power of three over (x power of seven y power of 3) and then power of 2

Answer 1

To simplify the expression \(\frac{(4x^4 y^{-4})^3}{(x^7 y^3)^2}\) using the properties of integer exponents, follow these steps:

1. Apply the Power of a Product Rule:

   • For the numerator, \((4x^4 y^{-4})^3\): \[ (4)^3(x^4)^3(y^{-4})^3 = 64x^{12}y^{-12} \]

   • For the denominator, \((x^7 y^3)^2\): \[ (x^7)^2(y^3)^2 = x^{14}y^6 \]

2. Combine the Numerator and Denominator: \[ \frac{64x^{12}y^{-12}}{x^{14}y^6} \]

3. Apply the Quotient Rule:

   • For \(x\): \[ x^{12-14} = x^{-2} \]

   • For \(y\): \[ y^{-12-6} = y^{-18} \]

4. Combine the results: \[ \frac{64}{x^2y^{18}} \]

So, the final equivalent expression is: \[ \frac{64}{x^2y^{18}} \]