Complete the Expression: (3×7)/(4×8) Raised to Power (b+1)

To solve the expression $\left(\frac{3\times7}{4\times8}\right)^{b+1}$, we follow these steps:

• Step 1: Apply the power $(b+1)$ to the entire fraction, $\left(\frac{3 \times 7}{4 \times 8}\right)^{b+1}$.
• Step 2: Use the exponentiation rule $(\frac{a}{b})^n = \frac{a^n}{b^n}$ to apply the power to both numerator and denominator separately.
• Step 3: Expand the numerator and the denominator: $(3 \times 7)^{b+1} = 3^{b+1} \times 7^{b+1}$ and $(4 \times 8)^{b+1} = 4^{b+1} \times 8^{b+1}$.
• Step 4: Combine these results into one fraction: $\frac{3^{b+1} \times 7^{b+1}}{4^{b+1} \times 8^{b+1}}$.

Therefore, the simplified expression is $\frac{3^{b+1}\times7^{b+1}}{4^{b+1}\times8^{b+1}}$.

Upon examining the choices, the correct option is choice 3: $\frac{3^{b+1}\times7^{b+1}}{4^{b+1}\times8^{b+1}}$.