Which of the following is equal to `y^((3)/(2))` for all values of y for which the expression is defined?

Options

A. `root(3)(y^(2))`
B. `sqrt(y^(3))`
C. `sqrt(y^((1)/(2)))`
D. `3sqrt(y)`

Answer

To solve the problem of finding which expression is equal to \( y^{\frac{3}{2}} \) for all values of \( y \) for which the expression is defined, we can follow these steps:

### Step 1: Rewrite the Expression
Start with the expression given in the question:
\[ y^{\frac{3}{2}} \]
This can be rewritten as:
\[ y^{\frac{3}{2}} = y^{3 \cdot \frac{1}{2}} = (y^3)^{\frac{1}{2}} \]

### Step 2: Apply the Property of Exponents
Using the property of exponents that states \( a^{m \cdot n} = (a^m)^n \), we can express the above as:
\[ (y^3)^{\frac{1}{2}} \]

### Step 3: Convert to Radical Form
The expression \( (y^3)^{\frac{1}{2}} \) can be converted to radical form. Recall that \( a^{\frac{1}{2}} \) is equivalent to \( \sqrt{a} \):
\[ (y^3)^{\frac{1}{2}} = \sqrt{y^3} \]

### Step 4: Identify the Correct Option
Now we can compare this result with the options provided:
1. \( \sqrt[3]{y^2} \)
2. \( \sqrt{y^3} \)
3. \( \sqrt{y^{\frac{1}{2}}} \)
4. \( \sqrt[3]{y} \)

From our derivation, we see that:
\[ y^{\frac{3}{2}} = \sqrt{y^3} \]
Thus, the correct option is:
\[ \text{Option 2: } \sqrt{y^3} \]

### Final Answer
The expression equal to \( y^{\frac{3}{2}} \) for all values of \( y \) for which the expression is defined is:
\[ \sqrt{y^3} \]

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