To simplify the given radical expression [tex]\(\frac{7}{\sqrt{3}}\)[/tex] by rationalizing the denominator, follow these steps:
1. Identify the expression: The given expression is [tex]\(\frac{7}{\sqrt{3}}\)[/tex].
2. Rationalize the denominator: To eliminate the radical in the denominator, multiply both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]. This technique works because [tex]\(\sqrt{3} \times \sqrt{3} = 3\)[/tex].
[tex]\[
\frac{7}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{7 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}
\][/tex]
3. Perform the multiplication:
- The numerator becomes: [tex]\(7 \cdot \sqrt{3} = 7\sqrt{3}\)[/tex]
- The denominator is simplified as: [tex]\(\sqrt{3} \cdot \sqrt{3} = 3\)[/tex]
Therefore, this gives us:
[tex]\[
\frac{7\sqrt{3}}{3}
\][/tex]
4. Simplify further if necessary: In this case, the fraction is already simplified. So, the final simplified form is:
[tex]\[
\frac{7\sqrt{3}}{3}
\][/tex]
5. Numerical approximation: If you require a numerical approximation of this expression, you can calculate it as follows:
- The value of [tex]\(\sqrt{3}\)[/tex] is approximately [tex]\(1.7320508075688772\)[/tex]
- Therefore, [tex]\(7 \cdot \sqrt{3} \approx 7 \cdot 1.7320508075688772 \approx 12.12435565298214\)[/tex]
- And the denominator is [tex]\(3\)[/tex]
Thus, the simplified radical expression in its decimal form is:
[tex]\[
\frac{12.12435565298214}{3} \approx 4.04145188432738
\][/tex]
But the simplified radical form remains [tex]\(\frac{7\sqrt{3}}{3}\)[/tex].
