Apply All Exponent Rules Practice

Question

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Question
Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13
with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13=

Remaining Attempts : 3

Answer 1

To simplify the expression \((4^3 \cdot 7^{-2})^{-5} / (7^9 \cdot 4^{-13})\), we will apply the rules of exponents step by step.

1. Apply the power of a product rule: \((a \cdot b)^n = a^n \cdot b^n\)

   \[ (4^3 \cdot 7^{-2})^{-5} = 4^{3 \cdot -5} \cdot 7^{-2 \cdot -5} = 4^{-15} \cdot 7^{10} \]

   So, we have: \[ \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} \]

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

3. Apply the quotient of powers rule: \(\frac{a^m}{a^n} = a^{m-n}\)

   For \(4\): \[ \frac{4^{-15}}{4^{-13}} = 4^{-15 - (-13)} = 4^{-15 + 13} = 4^{-2} \]

   For \(7\): \[ \frac{7^{10}}{7^{9}} = 7^{10 - 9} = 7^{1} = 7 \]

   So we have: \[ 4^{-2} \cdot 7 = \frac{7}{4^2} \]

4. Simplify further: \[ 4^2 = 16 \]

   Therefore: \[ \frac{7}{16} \]

So the final answer, simplified with only positive exponents, is: \[ \frac{7}{16} \]