Solve (3×7)/(5×8) Raised to Negative Third Power: Complex Fraction Challenge

Insert the corresponding expression:

$\left(\frac{3\times7}{5\times8}\right)^{-3}=$

The expression we are given is $\left(\frac{3\times7}{5\times8}\right)^{-3}$. In order to simplify it, we will apply the rules for negative exponents and powers of a fraction.

Step 1: Recognize that we are dealing with a negative exponent. The rule for negative exponents is $a^{-n} = \frac{1}{a^n}$. Thus, we invert the fraction and change the sign of the exponent:

$\left(\frac{3 \times 7}{5 \times 8}\right)^{-3} = \left(\frac{5 \times 8}{3 \times 7}\right)^{3}$

Step 2: Apply the power of a fraction rule, which states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$:

$\left(\frac{5 \times 8}{3 \times 7}\right)^{3} = \frac{(5 \times 8)^3}{(3 \times 7)^3}$

Step 3: Apply the power of a product rule, which allows us to distribute the exponent across the multiplication:

$\frac{(5)^3 \times (8)^3}{(3)^3 \times (7)^3} = \frac{5^3 \times 8^3}{3^3 \times 7^3}$

Step 4: Express each base raised to the power of -3 directly:

$\frac{5^{-3} \times 8^{-3}}{3^{-3} \times 7^{-3}}$

Since the inverted version of the expression can also mean distributing -3 directly across the original fraction components, this can be rearranged as:

$\frac{3^{-3} \times 7^{-3}}{5^{-3} \times 8^{-3}}$

Comparing with the given choices, the corresponding expression is choice 3:

$\frac{3^{-3}\times7^{-3}}{5^{-3}\times8^{-3}}$

Therefore, the equivalent expression for the given problem is $\frac{3^{-3}\times7^{-3}}{5^{-3}\times8^{-3}}$.