To rationalize the denominator and simplify the expression [tex]\(\frac{6}{\sqrt{3}}\)[/tex], follow these steps:
### Step 1: Multiply numerator and denominator by [tex]\(\sqrt{3}\)[/tex]
To rationalize the denominator, we need to eliminate the square root in the denominator. We do this by multiplying both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}
\][/tex]
### Step 2: Simplify the denominator
[tex]\(\sqrt{3} \cdot \sqrt{3}\)[/tex] equals 3, so the expression becomes:
[tex]\[
\frac{6 \cdot \sqrt{3}}{3}
\][/tex]
### Step 3: Multiply the numerator
Now, we perform the multiplication [tex]\(6 \cdot \sqrt{3}\)[/tex]:
[tex]\[
6 \cdot \sqrt{3} \approx 10.392304845413264
\][/tex]
### Step 4: Put this value over the rationalized denominator
Now, we have:
[tex]\[
\frac{10.392304845413264}{3}
\][/tex]
### Step 5: Simplify the fraction
Finally, simplify the fraction by performing the division:
[tex]\[
\frac{10.392304845413264}{3} \approx 3.4641016151377553
\][/tex]
So, the rationalized and simplified form of [tex]\(\frac{6}{\sqrt{3}}\)[/tex] is:
[tex]\[
\boxed{3.4641016151377553}
\][/tex]
