https://file.aiquickdraw.com/mathbot/file/1725442431743jee25932.jpg

The problem you're referring to in the image asks for the factorization of a radical expression, specifically the expression \(\left( 2^x - 2^{\frac{3x}{2}} \right)^{\frac{1}{2}}\).

To solve this, let's follow the steps:

### Given:
\[
\text{Expression to factorize: } \left( 2^x - 2^{\frac{3x}{2}} \right)^{\frac{1}{2}}
\]

### Step 1: Factor the expression inside the square root.
First, consider the expression inside the square root:
\[
2^x - 2^{\frac{3x}{2}}
\]

Factor out the common factor, which is \(2^x\):
\[
2^x(1 - 2^{\frac{x}{2}})
\]

### Step 2: Substitute the factored expression back under the square root.
Now, substitute the factored expression into the square root:
\[
\left( 2^x(1 - 2^{\frac{x}{2}}) \right)^{\frac{1}{2}}
\]

### Step 3: Simplify the expression.
Apply the square root to both factors:
\[
\sqrt{2^x} \cdot \sqrt{1 - 2^{\frac{x}{2}}}
\]

The square root of \(2^x\) is \(2^{\frac{x}{2}}\), so:
\[
2^{\frac{x}{2}} \cdot \sqrt{1 - 2^{\frac{x}{2}}}
\]

This is the simplified form of the given expression.

### Final Answer:
\[
\boxed{2^{\frac{x}{2}} \cdot \sqrt{1 - 2^{\frac{x}{2}}}}
\]

Let me know if you would like a detailed explanation or if you have any questions. Here are five related questions to expand your understanding:

1. How would you factor an expression like \(3^x - 3^{\frac{3x}{2}}\)?
2. What happens if the exponents in the expression are negative?
3. Can you generalize the factorization process for any base \(a\) instead of 2?
4. How would you deal with expressions that have more terms, like \(2^x - 2^{\frac{3x}{2}} + 2^{2x}\)?
5. How does the distributive property apply when factoring such expressions?

**Tip:** When dealing with expressions involving exponents, always check for a common base that can be factored out before applying other operations.

Rationalizing factor of \( \left( 2^x - 2^{\frac{3x}{2}} \right)^{\frac{1}{2}} \)

Explore the factorization and simplification of the radical expression \( \left( 2^x - 2^{\frac{3x}{2}} \right)^{\frac{1}{2}} \). This problem covers key algebraic concepts including exponentiation, radicals, and factorization. Suitable for 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