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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

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

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

x2y1864
Start Fraction x squared y superscript 18 baseline over 64 End Fraction

64x2y18

3 answers

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

  1. Simplify each part separately:

    \((4x^4y^{-4})^3 = 4^3 (x^4)^3 (y^{-4})^3\)

    • \(4^3 = 64\)
    • \((x^4)^3 = x^{12}\)
    • \((y^{-4})^3 = y^{-12}\)

    Therefore, \((4x^4y^{-4})^3 = 64x^{12}y^{-12}\).

  2. Now simplify the second part:

    \((x^7y^3)^2 = (x^7)^2 (y^3)^2\)

    • \((x^7)^2 = x^{14}\)
    • \((y^3)^2 = y^{6}\)

    Therefore, \((x^7y^3)^2 = x^{14}y^6\).

  3. Now, combine the two simplified parts:

    \[ (64x^{12}y^{-12})(x^{14}y^{6}) = 64x^{12+14}y^{-12+6} \]

    Simplifying the exponents:

    • For \(x\): \(12 + 14 = 26\)
    • For \(y\): \(-12 + 6 = -6\)

    This gives us: \[ 64x^{26}y^{-6} \]

  4. We can express \(y^{-6}\) as \(\frac{1}{y^6}\):

    \[ 64x^{26} \cdot \frac{1}{y^6} = \frac{64x^{26}}{y^6} \]

Thus, the simplified expression is:

\[ \frac{64x^{26}}{y^6} \]

None of the answer choices exactly match this result. However, if the expression options you provided included \(\frac{64x^{2}y^{18}}{1}\), that might suggest that perhaps there was a misunderstanding or error in the exponent calculations.

Please check the options against the expression we derived to find a correct match.

u sure

Let's go through the simplification step-by-step one more time to ensure we're clear and correct.

You started with the expression:

\[ (4x^4y^{-4})^3 (x^7y^3)^2 \]

Step 1: Simplify \((4x^4y^{-4})^3\)

Using the power of a product rule \((a^m b^n)^p = a^{mp} b^{np}\):

\[ (4x^4y^{-4})^3 = 4^3 (x^4)^3 (y^{-4})^3 \]

Calculating each part:

  • \(4^3 = 64\)
  • \((x^4)^3 = x^{12}\)
  • \((y^{-4})^3 = y^{-12}\)

Putting it all together:

\[ (4x^4y^{-4})^3 = 64x^{12}y^{-12} \]

Step 2: Simplify \((x^7y^3)^2\)

Using the same power of a product rule:

\[ (x^7y^3)^2 = (x^7)^2 (y^3)^2 \]

Calculating each part:

  • \((x^7)^2 = x^{14}\)
  • \((y^3)^2 = y^{6}\)

Putting it all together:

\[ (x^7y^3)^2 = x^{14}y^{6} \]

Step 3: Combine the results

Now combine the two results:

\[ (64x^{12}y^{-12})(x^{14}y^{6}) = 64x^{12 + 14}y^{-12 + 6} \]

Calculating the exponents:

  • For \(x\): \(12 + 14 = 26\)
  • For \(y\): \(-12 + 6 = -6\)

So we have:

\[ 64x^{26}y^{-6} \]

Step 4: Final expression

Express \(y^{-6}\) as \(\frac{1}{y^6}\):

\[ 64x^{26}y^{-6} = \frac{64x^{26}}{y^6} \]

Confirming the Result

The simplified expression is:

\[ \frac{64x^{26}}{y^6} \]

Unfortunately, neither of the original choices you provided includes this expression, suggesting that perhaps the options may not correctly reflect the computation we performed. If there are any specific matching criteria or constraints from the options you provided, please clarify, and I'll help accordingly!