https://file.aiquickdraw.com/mathbot/file/1726993019787bgtejbsz.jpg

Let's analyze the given mathematical expression:

\[
\frac{2^4 \cdot 2^3 \cdot 2^{-3}}{\sqrt{4^2 \cdot 2^4 \cdot 2^{-3}}} = 2^n
\]

We need to simplify this expression step by step to find the value of \( n \).

### Step 1: Simplifying the numerator
The numerator is:

\[
2^4 \cdot 2^3 \cdot 2^{-3}
\]

Using the property of exponents \( a^m \cdot a^n = a^{m+n} \), we combine the powers of 2:

\[
2^4 \cdot 2^3 \cdot 2^{-3} = 2^{4 + 3 - 3} = 2^4
\]

### Step 2: Simplifying the denominator
The denominator is the square root expression:

\[
\sqrt{4^2 \cdot 2^4 \cdot 2^{-3}}
\]

First, express \( 4^2 \) as \( (2^2)^2 = 2^4 \):

\[
\sqrt{2^4 \cdot 2^4 \cdot 2^{-3}} = \sqrt{2^{4+4-3}} = \sqrt{2^5}
\]

Now, simplify the square root of \( 2^5 \):

\[
\sqrt{2^5} = 2^{5/2}
\]

### Step 3: Putting it all together
Now the expression becomes:

\[
\frac{2^4}{2^{5/2}}
\]

Using the property \( \frac{a^m}{a^n} = a^{m-n} \):

\[
2^{4 - 5/2} = 2^{\frac{8}{2} - \frac{5}{2}} = 2^{\frac{3}{2}}
\]

Thus, we have:

\[
2^{n} = 2^{\frac{3}{2}}
\]

### Step 4: Conclusion
Therefore, \( n = \frac{3}{2} \).

---

Would you like a detailed explanation of any specific step or have further questions?

Here are 5 related questions for practice:
1. How do you simplify the product of exponents with the same base?
2. What is the square root of an expression involving exponents?
3. How would you approach a similar problem if the base was different from 2?
4. Can you simplify \( \frac{a^m}{a^n} \) for general values of \( m \) and \( n \)?
5. How does the fractional exponent \( a^{1/n} \) relate to roots?

**Tip**: When simplifying expressions with exponents, always combine terms with the same base using the properties of exponents.

The expression (2^4 * 2^3 * 2^-3) / √(4^2 * 2^4 * 2^-3) equals 2^n, where n is:

This problem demonstrates how to simplify an exponential expression involving multiplication, division, and square roots. The solution explains step-by-step methods to handle exponents and fractional powers, suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation