Question

Apply the Properties of Integer Exponents to solve the expression (7^−3)^7⋅7^20/6^−2
. Your answer should be an improper fraction with no exponents.(1 point)
=

Answer 1

Let's simplify the expression step-by-step using the properties of integer exponents.

The expression is:

\[
\frac{(7^{-3})^7 \cdot 7^{20}}{6^{-2}}
\]

### Step 1: Simplify \((7^{-3})^7\)

We use the power of a power property, which states that \((a^m)^n = a^{m \cdot n}\):

\[
(7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21}
\]

### Step 2: Combine \(7^{-21}\) and \(7^{20}\)

Now, we rewrite the expression as:

\[
\frac{7^{-21} \cdot 7^{20}}{6^{-2}}
\]

Next, we can combine the terms in the numerator by using the product of powers property, which states that \(a^m \cdot a^n = a^{m + n}\):

\[
7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1}
\]

### Step 3: Rewrite the expression

Now, we have:

\[
\frac{7^{-1}}{6^{-2}}
\]

### Step 4: Simplify \(6^{-2}\)

Next, by using the negative exponent property, \(a^{-m} = \frac{1}{a^m}\):

\[
6^{-2} = \frac{1}{6^2} = \frac{1}{36}
\]

### Step 5: Rewrite the fraction

We can rewrite the entire expression as follows:

\[
\frac{7^{-1}}{6^{-2}} = \frac{7^{-1}}{\frac{1}{36}} = 7^{-1} \cdot 36
\]

### Step 6: Apply the negative exponent property again

Now, we convert \(7^{-1}\):

\[
7^{-1} = \frac{1}{7}
\]

Thus, we have:

\[
\frac{1}{7} \cdot 36 = \frac{36}{7}
\]

### Final Answer

Hence, the final answer is:

\[
\boxed{\frac{36}{7}}
\]