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Apply the properties of integer exponents to generate an equivalent algebraic expression to (4x4y−4)3(x7y3)2
.(1 point)
Responses

64x20
Start Fraction 64 over x superscript 20 baseline End Fraction

4x2y18
Start Fraction 4 over x squared y superscript 18 baseline End Fraction

64x2y18
Start Fraction 64 over x squared y superscript 18 baseline End Fraction

x2y1864

1 answer

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

  1. Apply the power to each part of the product: \[ (4x^4y^{-4})^3 = 4^3 (x^4)^3 (y^{-4})^3 \] \[ (x^7y^3)^2 = (x^7)^2 (y^3)^2 \]

  2. Calculate each term: \[ 4^3 = 64 \] \[ (x^4)^3 = x^{4 \cdot 3} = x^{12} \] \[ (y^{-4})^3 = y^{-12} \] \[ (x^7)^2 = x^{7 \cdot 2} = x^{14} \] \[ (y^3)^2 = y^{3 \cdot 2} = y^6 \]

  3. Combine these results: \[ (4x^4y^{-4})^3 = 64x^{12}y^{-12} \] \[ (x^7y^3)^2 = x^{14}y^6 \]

  4. Combine the expressions: \[ 64x^{12}y^{-12} \cdot x^{14}y^6 = 64x^{12+14}y^{-12+6} \] \[ = 64x^{26}y^{-6} \]

  5. Rewrite in a simplified form: Since \(y^{-6} = \frac{1}{y^6}\), the expression can be rewritten as: \[ = \frac{64x^{26}}{y^6} \]

Now let's look at your options:

  1. \( \frac{64}{x^{20}} \)
  2. \( \frac{4}{x^2y^{18}} \)
  3. \( \frac{64}{x^2y^{18}} \)
  4. \( x^2y^{1864} \)

None of these responses match our calculated expression \(\frac{64x^{26}}{y^6}\).

However, there seems to be an error in the options as they do not correspond to the correct simplification of \((4x^4y^{-4})^3(x^7y^3)^2\). The correct result is: \[ \frac{64x^{26}}{y^{6}} \]