To rationalize the denominator and simplify the expression [tex]\( \sqrt{\frac{10}{14}} \)[/tex], follow these detailed steps:
1. Simplify the Fraction Inside the Square Root:
- The given expression is [tex]\( \sqrt{\frac{10}{14}} \)[/tex].
- First, simplify the fraction [tex]\(\frac{10}{14}\)[/tex]. The greatest common divisor (GCD) of 10 and 14 is 2.
- Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{10 \div 2}{14 \div 2} = \frac{5}{7}
\][/tex]
- So, [tex]\( \sqrt{\frac{10}{14}} \)[/tex] simplifies to [tex]\( \sqrt{\frac{5}{7}} \)[/tex].
2. Express [tex]\( \sqrt{\frac{5}{7}} \)[/tex] as a Division of Two Square Roots:
[tex]\[
\sqrt{\frac{5}{7}} = \frac{\sqrt{5}}{\sqrt{7}}
\][/tex]
3. Rationalize the Denominator:
- To rationalize the denominator, multiply both the numerator and the denominator by [tex]\( \sqrt{7} \)[/tex]:
[tex]\[
\frac{\sqrt{5}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{5} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}}
\][/tex]
4. Simplify the Expression:
- The numerator becomes [tex]\( \sqrt{5} \cdot \sqrt{7} = \sqrt{35} \)[/tex].
- The denominator becomes [tex]\( \sqrt{7} \cdot \sqrt{7} = 7 \)[/tex].
- Hence, the expression simplifies to:
[tex]\[
\frac{\sqrt{35}}{7}
\][/tex]
So, the rationalized and simplified form of [tex]\( \sqrt{\frac{10}{14}} \)[/tex] is [tex]\( \frac{\sqrt{35}}{7} \)[/tex].
### Additional Numerical Details:
- The GCD of 10 and 14 is 2.
- The simplified fraction is [tex]\( \frac{5}{7} \)[/tex].
- Square root of 5: [tex]\( \sqrt{5} \approx 2.236 \)[/tex].
- Square root of 7: [tex]\( \sqrt{7} \approx 2.646 \)[/tex].
- Rationalized numerator: [tex]\( \sqrt{35} \approx 5.916 \)[/tex].
- Rationalized denominator: [tex]\( 7 \)[/tex].
Thus, the final rationalized form [tex]\( \frac{\sqrt{35}}{7} \approx \frac{5.916}{7} \)[/tex].
