Question

Question
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

Answer 1

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