To simplify the radical expression [tex]\( \frac{5}{\sqrt{7}} \)[/tex] by rationalizing the denominator, follow these steps:
1. Multiply the numerator and the denominator by [tex]\( \sqrt{7} \)[/tex]
The goal of rationalizing the denominator is to eliminate the square root from the denominator. To do this, you multiply both the numerator and denominator by [tex]\( \sqrt{7} \)[/tex]:
[tex]\[
\frac{5}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}
\][/tex]
2. Multiply the terms in the numerator and the denominator separately
First, multiply the numerators:
[tex]\[
5 \times \sqrt{7} = 5\sqrt{7}
\][/tex]
Next, multiply the denominators:
[tex]\[
\sqrt{7} \times \sqrt{7} = 7
\][/tex]
3. Combine the new numerator and denominator
Putting everything together, we get:
[tex]\[
\frac{5 \sqrt{7}}{7}
\][/tex]
4. Verify the simplified radical expression
To confirm the expression is fully simplified, we see that the denominator is now a rational number (7), and there are no radicals left in the denominator. The numerator remains [tex]\( 5 \sqrt{7} \)[/tex], which cannot be simplified further.
Therefore, the simplified and rationalized form of [tex]\( \frac{5}{\sqrt{7}} \)[/tex] is:
[tex]\[
\frac{5 \sqrt{7}}{7}
\][/tex]
Given the numerical values obtained from this rationalization process:
- The rationalized numerator [tex]\( 5 \sqrt{7} \)[/tex] is approximately [tex]\( 13.228756555322953 \)[/tex]
- The rationalized denominator is [tex]\( 7 \)[/tex]
- Consequently, the simplified fraction is approximately [tex]\( 1.8898223650461359 \)[/tex]
