Question

Apply the properties of integer exponents to generate equivalent expressions to in parentheses four with the power of 3×7 with a negative power to and on the outside of the parentheses we have a -5 power over seven with a power of 9×4 with the power of -13 with only positive exponents then solve the expression your answer will be a simplified fraction with no remaining exponents

Answer 1

To simplify the expression \( \left( 4^{3 \times 7} \right)^{-5} \div \left( 7^{9 \times 4} \right)^{-13} \) using the properties of integer exponents, we'll follow these steps:

1. **Calculate the exponents:**
\( 3 \times 7 = 21 \)
\( 9 \times 4 = 36 \)

Therefore, we can rewrite the expression as:
\[
\left( 4^{21} \right)^{-5} \div \left( 7^{36} \right)^{-13}
\]

2. **Apply the negative exponent rule:**
Recall that \( a^{-m} = \frac{1}{a^{m}} \). Thus, we can rewrite the expression:
\[
\frac{1}{4^{21 \cdot 5}} \div \frac{1}{7^{36 \cdot 13}}
\]
This simplifies to:
\[
\frac{7^{36 \cdot 13}}{4^{21 \cdot 5}}
\]

3. **Calculate the new exponents from the multiplication:**
\( 21 \cdot 5 = 105 \)
\( 36 \cdot 13 = 468 \)

Now the expression is:
\[
\frac{7^{468}}{4^{105}}
\]

4. **Rewrite \( 4 \) in terms of \( 2 \):**
Since \( 4 = 2^2 \), we can express \( 4^{105} \) as \( (2^2)^{105} = 2^{2 \cdot 105} = 2^{210} \).

Therefore, we now have:
\[
\frac{7^{468}}{2^{210}}
\]

5. **Final expression:**
Thus the equivalent expression with no remaining exponents and fully simplified is:
\[
\frac{7^{468}}{2^{210}}
\]

This fraction cannot be simplified further since 7 and 2 are prime bases. The final answer is:
\[
\frac{7^{468}}{2^{210}}
\]