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 answer

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

First, we will simplify the numerator \((4^3 \cdot 7^{-2})^{-5}\):

  1. Apply the property \((a^m \cdot b^n)^p = a^{mp} \cdot b^{np}\): \[ (4^3 \cdot 7^{-2})^{-5} = 4^{3 \cdot (-5)} \cdot 7^{-2 \cdot (-5)} = 4^{-15} \cdot 7^{10} \]

Now let's rewrite the entire expression: \[ \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} \]

Next, we simplify the denominator \(7^9 \cdot 4^{-13}\): \[ = 7^9 \cdot 4^{-13} \]

Now, we can rewrite the fraction: \[ \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} = \frac{4^{-15}}{4^{-13}} \cdot \frac{7^{10}}{7^9} \]

Using the property \(\frac{a^m}{a^n} = a^{m-n}\):

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

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

Now we combine the results: \[ 4^{-2} \cdot 7 = \frac{7}{4^2} \]

Calculating \(4^2\): \[ 4^2 = 16 \]

Thus, we have: \[ \frac{7}{16} \]

The final answer, simplified and with no remaining exponents, is: \[ \boxed{\frac{7}{16}} \]