Sure! Let's simplify the expression [tex]\( \frac{3}{\sqrt{3}} \)[/tex] by expressing it with radicals.
### Step 1: Expressing the Division
We start with the division in the numerator and the denominator:
[tex]\[
\frac{3}{\sqrt{3}}
\][/tex]
### Step 2: Rationalizing the Denominator
To simplify the expression, we need to get rid of the square root in the denominator. We do this by multiplying both the numerator and the denominator by [tex]\( \sqrt{3} \)[/tex]:
[tex]\[
\frac{3}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}
\][/tex]
### Step 3: Performing the Multiplication
Let's multiply both the numerator and the denominator:
[tex]\[
\frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}
\][/tex]
### Step 4: Simplifying the Denominator
The denominator simplifies because [tex]\( \sqrt{3} \cdot \sqrt{3} = (\sqrt{3})^2 = 3 \)[/tex]:
[tex]\[
\frac{3 \cdot \sqrt{3}}{3}
\][/tex]
### Step 5: Simplifying the Fraction
Since [tex]\( \frac{3 \cdot \sqrt{3}}{3} = \frac{3}{3} \cdot \sqrt{3} \)[/tex]:
[tex]\[
\frac{3}{3} \cdot \sqrt{3} = 1 \cdot \sqrt{3} = \sqrt{3}
\][/tex]
### Conclusion
Therefore, the simplified form of [tex]\( \frac{3}{\sqrt{3}} \)[/tex] is:
[tex]\[
\sqrt{3}
\][/tex]
In decimal form, this value is approximately [tex]\( 1.7320508075688774 \)[/tex].
### Verification
We can verify the simplified answer by looking at the numerical division:
1. The result of the initial division:
[tex]\[
\frac{3}{\sqrt{3}} \approx 1.7320508075688774
\][/tex]
2. The simplified form in terms of radicals:
[tex]\[
\sqrt{3} \approx 1.7320508075688774
\][/tex]
Both represent approximately the same value, confirming our simplification is correct.
