Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 times 7^-2)^-5/7^9 times 4^-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 \((4^3 \cdot 7^{-2})^{-5} \cdot \frac{1}{7^9} \cdot 4^{-13}\) and rewrite it with only positive exponents, we will apply the properties of integer exponents step by step.

1. Distributing the exponent on \((4^3 \cdot 7^{-2})^{-5}\): \[ (4^3 \cdot 7^{-2})^{-5} = 4^{3 \cdot -5} \cdot 7^{-2 \cdot -5} = 4^{-15} \cdot 7^{10}. \]

2. Rewrite the entire expression: \[ 4^{-15} \cdot 7^{10} \cdot \frac{1}{7^9} \cdot 4^{-13}. \]

3. Convert \(\frac{1}{7^9}\) to a negative exponent: \[ 4^{-15} \cdot 7^{10} \cdot 7^{-9} \cdot 4^{-13}. \]

4. Combining the \(4\) terms: \[ 4^{-15} \cdot 4^{-13} = 4^{-15 + (-13)} = 4^{-28}. \]

5. Combining the \(7\) terms: \[ 7^{10} \cdot 7^{-9} = 7^{10 + (-9)} = 7^{1}. \]

6. Putting it all together: \[ 4^{-28} \cdot 7^{1} = \frac{7}{4^{28}}. \]

7. Rewrite the expression with positive exponents: \[ \frac{7}{4^{28}}. \]

The final answer is: \[ \frac{7}{4^{28}}. \]

Answer 2

thats wrong

Answer 3

Let's go through the steps carefully to simplify the expression \((4^3 \cdot 7^{-2})^{-5} \cdot \frac{1}{7^9} \cdot 4^{-13}\) correctly and ensure we have only positive exponents at the end.

Step 1: Simplify \((4^3 \cdot 7^{-2})^{-5}\)

Using the properties of exponents: \[ (4^3 \cdot 7^{-2})^{-5} = (4^3)^{-5} \cdot (7^{-2})^{-5} = 4^{-15} \cdot 7^{10} \]

Step 2: Combine with the rest of the expression

Now replace \(\frac{1}{7^9}\) with \(7^{-9}\): \[ 4^{-15} \cdot 7^{10} \cdot 7^{-9} \cdot 4^{-13} \]

Step 3: Combine like terms for the bases

Combining the \(4\) terms: \[ 4^{-15} \cdot 4^{-13} = 4^{-15 - 13} = 4^{-28} \]

Combining the \(7\) terms: \[ 7^{10} \cdot 7^{-9} = 7^{10 - 9} = 7^{1} \]

Step 4: Putting it all together

The combined expression becomes: \[ 4^{-28} \cdot 7^{1} = 7 \cdot 4^{-28} \]

Step 5: Rewrite with positive exponents

To write \(4^{-28}\) with a positive exponent: \[ 7 \cdot \frac{1}{4^{28}} = \frac{7}{4^{28}} \]

Final Answer

The final simplified expression is: \[ \frac{7}{4^{28}} \]

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