Apply the Properties of Integer Exponents to generate equivalent expressions to (4 to the power of 3 ⋅ 7 to the power of −2) to the power of −5 / 7 to the power of 9 ⋅ 4 to the power of −13

Question

with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(43 ⋅ 7−2)−579 ⋅ 4−13=

Answer 1

To simplify the expression \((4^3 \cdot 7^{-2})^{-5} / (7^9 \cdot 4^{-13})\), we'll start by applying the properties of integer exponents.

Step 1: Simplify the Numerator

We begin with the numerator \((4^3 \cdot 7^{-2})^{-5}\): \[ (4^3 \cdot 7^{-2})^{-5} = (4^3)^{-5} \cdot (7^{-2})^{-5} \] Using the property \((a^m)^n = a^{m \cdot n}\): \[ = 4^{-15} \cdot 7^{10} \]

Step 2: Rewrite the Expression

Now substituting back into the expression, we have: \[ \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} \]

Step 3: Apply the Quotient Rule

Using the quotient rule of exponents, \(a^m / a^n = a^{m - n}\): \[ = \frac{4^{-15}}{4^{-13}} \cdot \frac{7^{10}}{7^9} \] Calculating each term: \[ = 4^{-15 - (-13)} \cdot 7^{10 - 9} \] This simplifies to: \[ = 4^{-2} \cdot 7^1 \]

Step 4: Rewrite with Positive Exponents

To express \(4^{-2}\) with positive exponents: \[ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \] Thus, we rewrite our expression: \[ = \frac{1}{16} \cdot 7 = \frac{7}{16} \]

Final Answer

So, the simplified fraction with no remaining exponents is: \[ \boxed{\frac{7}{16}} \]