Adding and Subtracting Powers - YouTube - Free Printable

Problem:

Simplify the expression:
\[
(2^{1/2})^5 - (2^{1/2})^3
\]

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Step-by-Step Solution:

#### Step 1: Simplify each term using the power rule
The power rule for exponents states that \((a^m)^n = a^{m \cdot n}\). We will apply this rule to both terms in the expression.

1. Simplify \((2^{1/2})^5\):
\[
(2^{1/2})^5 = 2^{(1/2) \cdot 5} = 2^{5/2}
\]

2. Simplify \((2^{1/2})^3\):
\[
(2^{1/2})^3 = 2^{(1/2) \cdot 3} = 2^{3/2}
\]

So, the expression becomes:
\[
2^{5/2} - 2^{3/2}
\]

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#### Step 2: Factor out the common term
Notice that both terms in the expression \(2^{5/2} - 2^{3/2}\) have a common factor of \(2^{3/2}\). We can factor this out:

\[
2^{5/2} - 2^{3/2} = 2^{3/2} \cdot 2^{(5/2 - 3/2)} - 2^{3/2} \cdot 1
\]

Simplify the exponent in the first term:
\[
5/2 - 3/2 = 2/2 = 1
\]

So, the expression becomes:
\[
2^{5/2} - 2^{3/2} = 2^{3/2} \cdot 2^1 - 2^{3/2} \cdot 1
\]

Using the property \(2^1 = 2\), this simplifies to:
\[
2^{5/2} - 2^{3/2} = 2^{3/2} \cdot 2 - 2^{3/2} \cdot 1
\]

Factor out \(2^{3/2}\):
\[
2^{5/2} - 2^{3/2} = 2^{3/2} (2 - 1)
\]

Simplify the expression inside the parentheses:
\[
2 - 1 = 1
\]

So, the expression becomes:
\[
2^{5/2} - 2^{3/2} = 2^{3/2} \cdot 1 = 2^{3/2}
\]

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#### Step 3: Write the final answer
The simplified form of the expression is:
\[
\boxed{2^{3/2}}
\]

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Verification:

To verify, recall that \(2^{3/2} = (2^3)^{1/2} = 8^{1/2} = \sqrt{8} = 2\sqrt{2}\). This matches the structure of the original expression after simplification.

Thus, the final answer is:
\[
\boxed{2^{3/2}}
\]