Simplify the Expression: Division of Powers b^11 ÷ b^8

To solve this problem, we will apply the Quotient Rule for exponents, which helps simplify expressions where both the numerator and denominator share the same base.

• Step 1: Identify the given information.
  We are given the expression $\frac{b^{11}}{b^8}$.
• Step 2: Apply the Quotient Rule for exponents.
  According to the Quotient Rule, $\frac{b^m}{b^n} = b^{m-n}$. Here, $m = 11$ and $n = 8$.
• Step 3: Subtract the exponent of the denominator from the exponent of the numerator.
  Calculate $11 - 8 = 3$, leading to the simplified expression $b^3$.

Therefore, the simplified form of the given expression is $b^{3}$.

When considering the choices:

• Choice 1: $b^{11+8}$ adds the exponents, which is incorrect for division.
• Choice 2: $b^{11\times8}$ multiplies the exponents, also incorrect for division.
• Choice 3: $b^{\frac{11}{8}}$ divides the exponents, which is incorrect for division.
• Choice 4: $b^{11-8}$, correctly subtracts the exponents as per the Quotient Rule.

Thus, the correct choice is Choice 4: $b^{11-8}$, which simplifies to $b^3$.