To simplify the expression \( b^{-b^{b^{-b} \cdot (b + \frac{1}{2})}} \), we'll break it down step by step.

### Step 1: Simplify the Exponent

The expression inside the exponent is \( -b^{b^{-b} \cdot (b + \frac{1}{2})} \).

Let's simplify the exponent part by part:

1. **Simplify \( b^{-b} \):**
   \[
   b^{-b} = \frac{1}{b^b}
   \]

2. **Multiply \( b^{-b} \) by \( (b + \frac{1}{2}) \):**
   \[
   b^{-b} \cdot (b + \frac{1}{2}) = \frac{1}{b^b} \cdot \left(b + \frac{1}{2}\right) = \frac{b + \frac{1}{2}}{b^b}
   \]

Thus, the exponent simplifies to:

\[
-b^{\frac{b + \frac{1}{2}}{b^b}}
\]

### Step 2: Rewriting the Expression

Now, substitute the simplified exponent back into the original expression:

\[
b^{-b^{\frac{b + \frac{1}{2}}{b^b}}}
\]

### Step 3: Simplify the Final Expression

The exponent \( -b^{\frac{b + \frac{1}{2}}{b^b}} \) is complex and not easily simplified further without specific values for \( b \). However, the structure of the expression indicates that \( b \) raised to a negative exponent will result in the reciprocal of \( b \) raised to the positive exponent:

\[
b^{-b^{\frac{b + \frac{1}{2}}{b^b}}} = \frac{1}{b^{b^{\frac{b + \frac{1}{2}}{b^b}}}}
\]

### Final Simplified Expression

The simplified expression is:

\[
\frac{1}{b^{b^{\frac{b + \frac{1}{2}}{b^b}}}}
\]

This is the most simplified form of the expression.

Would you like to explore this further or have any other questions?

Here are some related questions for further practice:

1. What happens if \( b = 1 \) in the original expression?
2. How would the expression change if the exponent were \( b^{-b^2 \cdot (b + \frac{1}{2})} \)?
3. What if the original expression had a coefficient in front of \( b \)?
4. How can you simplify the expression for specific values of \( b \)?
5. What are the properties of exponential functions when dealing with complex exponents?

**Tip:** When dealing with complex exponents, it's often helpful to rewrite the exponents in terms of fractions or simpler expressions to better understand the structure.

Learn how to simplify the exponential expression b^{-b^{b^{-b} * (b + 1/2)}} step-by-step with a detailed solution. This problem covers exponentiation, negative exponents, and fractional exponents, suitable for advanced high school and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation