Certainly! Let's go through the steps to rationalize the denominator of the fraction [tex]\(\frac{15}{\sqrt{3}}\)[/tex] in detail.
1. Understand the Goal: Rationalizing the denominator means we want to eliminate the square root in the denominator.
2. Multiply by a Form of 1: To eliminate the square root in the denominator, we can multiply both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex], because any number divided by itself equals 1 and it does not change the value of the fraction.
So, [tex]\(\frac{15}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{15 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}\)[/tex].
3. Simplify the Denominator: Simplify the denominator by recognizing that [tex]\(\sqrt{3} \cdot \sqrt{3}\)[/tex] equals 3.
Thus, [tex]\(\frac{15 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{15 \cdot \sqrt{3}}{3}\)[/tex].
4. Simplify the Numerator: The numerator remains [tex]\(15 \cdot \sqrt{3}\)[/tex].
5. Divide the Constants: Now, divide the constant terms (if possible):
[tex]\( \frac{15 \cdot \sqrt{3}}{3} = 5 \cdot \sqrt{3} \)[/tex].
So, the fraction [tex]\(\frac{15}{\sqrt{3}}\)[/tex] with a rationalized denominator is [tex]\(5 \cdot \sqrt{3}\)[/tex].
### Numerical Value
To express this in terms of a numerical value:
- Calculate [tex]\( \sqrt{3} \approx 1.732 \)[/tex].
- Thus, [tex]\( 5 \cdot \sqrt{3} \approx 5 \cdot 1.732 = 8.660 \)[/tex].
Therefore, the numerical value of the fraction [tex]\(\frac{15}{\sqrt{3}}\)[/tex] when rationalized is approximately [tex]\(8.660\)[/tex].
In conclusion:
- The simplified form of [tex]\(\frac{15}{\sqrt{3}}\)[/tex] is [tex]\(5 \sqrt{3}\)[/tex].
- The numerical value of this simplified fraction is approximately [tex]\(8.660\)[/tex].
