Complete the Expression: (a⁵x⁵)/(7⁵b⁵) Algebraic Fraction

To solve this problem, our goal is to express the given quotient of powers in a simplified form using exponent laws.

• Step 1: Understand the original expression. We have $\frac{a^5 \times x^5}{7^5 \times b^5}$.
• Step 2: Recognize the structure. Notice that both the numerator and denominator are raised to the fifth power.
• Step 3: Apply the property of exponents for quotients and products, which states that $\left(\frac{m}{n}\right)^k = \frac{m^k}{n^k}$ and $(m \cdot n)^k = m^k \cdot n^k$.
• Step 4: Rewrite the expression as a single fraction raised to the power of 5. Since each term in the numerator and denominator is raised to the fifth power separately, we combine them under a single power:
• Numerator: $a^5 \times x^5 = (a \times x)^5$
• Denominator: $7^5 \times b^5 = (7 \times b)^5$
• Therefore, $\frac{a^5 \times x^5}{7^5 \times b^5} = \left(\frac{a \times x}{7 \times b}\right)^5$.

Thus, the expression can be written as: $\left(\frac{a \times x}{7 \times b}\right)^5$.

Now, comparing this with the answer choices provided:

• Choice 1: $\frac{(a \times x)^5}{7 \times b^5}$ - does not match, as it retains the separate powers incorrectly.
• Choice 2: $\left(\frac{a \times x}{7 \times b}\right)^5$ - matches perfectly as derived.
• Choice 3: $\frac{a \times x^5}{(7 \times b)^5}$ - incorrect form compared to derived structure.
• Choice 4: $7 \times \left(\frac{a \times x}{b}\right)^5$ - unrelated format, doesn't match.

The correct choice is therefore Choice 2. This matches our derived expression using the laws of exponents correctly.