To evaluate the expression [tex]\(\frac{\sqrt{2} - 1}{\sqrt{2} + 1} + \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}\)[/tex] correct to three decimal places, follow these detailed steps:
1. Assign the approximate values:
[tex]\[
\sqrt{2} = 1.414
\][/tex]
[tex]\[
\sqrt{3} = 1.732
\][/tex]
2. Evaluate the first fraction [tex]\(\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\)[/tex]:
- Calculate the numerator:
[tex]\[
\sqrt{2} - 1 = 1.414 - 1 = 0.414
\][/tex]
- Calculate the denominator:
[tex]\[
\sqrt{2} + 1 = 1.414 + 1 = 2.414
\][/tex]
- Compute the fraction:
[tex]\[
\frac{0.414}{2.414} \approx 0.171
\][/tex]
3. Evaluate the second fraction [tex]\(\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}\)[/tex]:
- Calculate the numerator:
[tex]\[
\sqrt{3} - \sqrt{2} = 1.732 - 1.414 = 0.318
\][/tex]
- Calculate the denominator:
[tex]\[
\sqrt{3} + \sqrt{2} = 1.732 + 1.414 = 3.146
\][/tex]
- Compute the fraction:
[tex]\[
\frac{0.318}{3.146} \approx 0.101
\][/tex]
4. Add the two fractions together:
[tex]\[
0.171 + 0.101 = 0.272
\][/tex]
5. Round the result to three decimal places:
[tex]\[
\boxed{0.273}
\][/tex]
Therefore, the value of the expression [tex]\(\frac{\sqrt{2} - 1}{\sqrt{2} + 1} + \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}\)[/tex] correct to three decimal places is [tex]\(0.273\)[/tex].
