`((b^(2n+1))^(3))/(b^(n)*b^(5n+1))`
The expression above is equivalent to which of the following?

Options

A. `b^(2)`
B. `(1)/(b^(2))`
C. `b^(2n)`
D. `(1)/(b^(2n))`

Answer

To solve the expression \(\frac{(b^{2n+1})^3}{b^n \cdot b^{5n+1}}\), we will follow these steps:

### Step 1: Simplify the Numerator
The numerator is \((b^{2n+1})^3\). When we raise a power to another power, we multiply the exponents.

\[ (b^{2n+1})^3 = b^{(2n+1) \cdot 3} = b^{6n + 3} \]

### Step 2: Simplify the Denominator
The denominator is \(b^n \cdot b^{5n+1}\). When multiplying powers with the same base, we add the exponents.

\[ b^n \cdot b^{5n+1} = b^{n + (5n + 1)} = b^{6n + 1} \]

### Step 3: Combine the Numerator and Denominator
Now we can rewrite the expression as:

\[ \frac{b^{6n + 3}}{b^{6n + 1}} \]

### Step 4: Simplify the Fraction
When dividing powers with the same base, we subtract the exponents:

\[ b^{(6n + 3) - (6n + 1)} = b^{6n + 3 - 6n - 1} = b^{2} \]

### Conclusion
Thus, the expression simplifies to:

\[ b^2 \]

### Final Answer
The expression \(\frac{(b^{2n+1})^3}{b^n \cdot b^{5n+1}}\) is equivalent to \(b^2\).

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