The expression cube root of, 2, to the power 5 , end cube root, dot, square root of, 2, cubed, end square root 
3
  
2 
5
 
​
 ⋅ 
2 
3
 
​
  is equivalent to

Let's simplify the given expression step by step. The expression is:

\[
\sqrt[3]{2^5} \cdot \sqrt{2^3}
\]

### Step 1: Simplify the cube root term
The cube root of \(2^5\) can be written as:
\[
\sqrt[3]{2^5} = 2^{\frac{5}{3}}
\]

### Step 2: Simplify the square root term
The square root of \(2^3\) can be written as:
\[
\sqrt{2^3} = 2^{\frac{3}{2}}
\]

### Step 3: Multiply the simplified expressions
Now we have:
\[
2^{\frac{5}{3}} \cdot 2^{\frac{3}{2}}
\]

When multiplying powers with the same base, we add the exponents:
\[
2^{\frac{5}{3} + \frac{3}{2}}
\]

### Step 4: Add the exponents
To add \(\frac{5}{3}\) and \(\frac{3}{2}\), we find a common denominator:
\[
\frac{5}{3} = \frac{10}{6}, \quad \frac{3}{2} = \frac{9}{6}
\]
\[
\frac{10}{6} + \frac{9}{6} = \frac{19}{6}
\]

### Final Result
The expression simplifies to:
\[
2^{\frac{19}{6}}
\]

So, the equivalent form of the original expression is \(2^{\frac{19}{6}}\).

Would you like further details or have any questions?

### Related Questions
1. How do you simplify expressions with different roots (e.g., fourth roots)?
2. What are the general rules for adding and multiplying exponents?
3. How can you express a root as a fractional exponent?
4. What are the properties of cube roots and square roots in algebra?
5. How do you simplify complex expressions involving both roots and powers?

### Tip
When dealing with powers and roots, remember that any root can be expressed as a fractional exponent: \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\).

The expression cube root of, 2, to the power 5 , end cube root, dot, square root of, 2, cubed, end square root is equivalent to

Learn how to simplify the expression involving cube roots and square roots: cube root of 2^5 multiplied by square root of 2^3. This problem explores key algebraic concepts, including the properties of exponents and roots. Suitable for high school students (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