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 $2^{\frac{19}{6}}$.

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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: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.