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

Answer 1

To simplify the expression

\[ \frac{(4^3 \cdot 7^{-2})^{-5}}{7^9 \cdot 4^{-13}} \]

we will use properties of exponents. First, we will simplify the numerator and the denominator separately.

Step 1: Simplify the Numerator

The numerator is \((4^3 \cdot 7^{-2})^{-5}\). We apply the power of a product property: \((a \cdot b)^n = a^n \cdot b^n\).

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

Using the power of a power property: \((a^m)^n = a^{m \cdot n}\):

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

So the numerator simplifies to:

\[ 4^{-15} \cdot 7^{10} \]

Step 2: Simplify the Denominator

The denominator is \(7^9 \cdot 4^{-13}\). We keep it as it is for now because we'll deal with it when we combine the expression.

Now putting it together, we rewrite the full expression:

\[ \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} \]

Step 3: Combine the Fraction

Now we can combine the fractions by subtracting the exponents:

\[ = \frac{4^{-15}}{4^{-13}} \cdot \frac{7^{10}}{7^9} \]

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

\[ = 4^{-15 - (-13)} \cdot 7^{10 - 9} \]

Calculating the exponents:

\[ = 4^{-15 + 13} \cdot 7^{1} \]

So, we have:

\[ = 4^{-2} \cdot 7^{1} \]

Step 4: Change to Positive Exponents

Now we express \(4^{-2}\) as a positive exponent:

\[ = \frac{7^{1}}{4^{2}} = \frac{7}{4^2} \]

Calculating \(4^2\):

\[ = \frac{7}{16} \]

Final Answer

Thus, the final simplified expression is:

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