Use the Negative Exponent Rule to generate an equivalent expression to 3 -1/2

The Negative Exponent Rule states that \( a^{-n} = \frac{1}{a^n} \).

To use this rule on the expression \( 3^{-1/2} \), we can rewrite it as:

\[
3^{-1/2} = \frac{1}{3^{1/2}}
\]

Since \( 3^{1/2} \) is the square root of 3, we can also express the equivalent expression as:

\[
3^{-1/2} = \frac{1}{\sqrt{3}}
\]

Thus, an equivalent expression to \( 3^{-1/2} \) is \( \frac{1}{\sqrt{3}} \).

can you explain in another way?

Sure! Let's break it down step by step using a more intuitive approach.

### Step 1: Understanding Negative Exponents

When we see a negative exponent, it means we are dealing with a reciprocal. Specifically, the expression \( a^{-n} \) is telling us to take the reciprocal of \( a^n \).

### Step 2: Apply the Rule

So for your expression \( 3^{-1/2} \):
- The base is \( 3 \).
- The exponent is \( -\frac{1}{2} \).

Applying the Negative Exponent Rule, we rewrite \( 3^{-1/2} \) as:

\[
3^{-1/2} = \frac{1}{3^{1/2}}
\]

### Step 3: Simplify Further

Now, we need to understand \( 3^{1/2} \). The exponent \( \frac{1}{2} \) represents the square root. In simple terms:

- \( 3^{1/2} \) is the same as \( \sqrt{3} \).

So now we substitute \( 3^{1/2} \) with \( \sqrt{3} \):

\[
3^{-1/2} = \frac{1}{\sqrt{3}}
\]

### Final Result

Thus, in a straightforward manner, we have shown that:

\[
3^{-1/2} = \frac{1}{\sqrt{3}}
\]

We used the understanding of negative exponents to express it as a fraction, and then we recognized that \( 3^{1/2} \) corresponds to the square root of 3. This gives us the equivalent expression.

I'm still confused what the answer is?

I apologize for any confusion! To clarify, the equivalent expression for \( 3^{-1/2} \) is:

\[
\frac{1}{\sqrt{3}}
\]

So, to summarize:

- **Original expression**: \( 3^{-1/2} \)
- **Equivalent expression**: \( \frac{1}{\sqrt{3}} \)

If you have any more questions or need further clarification, feel free to ask!