Question

Apply the properties of integer exponents to generate equivalent expressions to (7^-3)^7X7^20/6^-2 with positive exponents then solve the expression

Answer 1

To simplify the expression \((7^{-3})^7 \times 7^{20} / 6^{-2}\) and express it with positive exponents, we'll use the properties of integer exponents.

1. **Apply the power of a power rule**: \((a^m)^n = a^{m \cdot n}\).
\[
(7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21}
\]

2. **Combine the terms with the same base**: We now have \(7^{-21} \times 7^{20}\).
Using the property \(a^m \times a^n = a^{m+n}\):
\[
7^{-21} \times 7^{20} = 7^{-21 + 20} = 7^{-1}
\]

3. **Convert the exponent to positive**: Remember that \(a^{-n} = \frac{1}{a^n}\):
\[
7^{-1} = \frac{1}{7}
\]

4. **Simplify the denominator**: Now we simplify \(6^{-2}\) as well:
\[
6^{-2} = \frac{1}{6^2} = \frac{1}{36}
\]

5. **Combine everything into a single expression**:
\[
\frac{1}{7} \div \frac{1}{36}
\]

6. **Dividing by a fraction is the same as multiplying by its reciprocal**:
\[
\frac{1}{7} \times 36 = \frac{36}{7}
\]

Thus, the expression \((7^{-3})^7 \times 7^{20} / 6^{-2}\) simplifies and evaluates to:
\[
\frac{36}{7}
\]