Expression Evaluation: Simplifying Fractions with Square Roots
The problem asks for the approximate value of the sum \(\frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}}\). We can simplify this expression by rationalizing the denominators of each fraction.
Rationalizing the First Term
Consider the first term: \(\frac{1}{1+\sqrt{2}}\).
- Multiply the numerator and the denominator by the conjugate of the denominator, which is \(1-\sqrt{2}\): \(\frac{1}{1+\sqrt{2}} \times \frac{1-\sqrt{2}}{1-\sqrt{2}}\)
- Apply the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) to the denominator: \(\frac{1-\sqrt{2}}{1^2 - (\sqrt{2})^2} = \frac{1-\sqrt{2}}{1-2} = \frac{1-\sqrt{2}}{-1}\)
- Simplify the fraction: \(\sqrt{2} - 1\)
Rationalizing the Second Term
Consider the second term: \(\frac{1}{\sqrt{2}+\sqrt{3}}\). It's easier to write the denominator as \(\sqrt{3}+\sqrt{2}\).
- Multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{3}-\sqrt{2}\): \(\frac{1}{\sqrt{3}+\sqrt{2}} \times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}\)
- Apply the difference of squares formula to the denominator: \(\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{\sqrt{3}-\sqrt{2}}{3-2} = \frac{\sqrt{3}-\sqrt{2}}{1}\)
- Simplify the fraction: \(\sqrt{3} - \sqrt{2}\)
Summing the Simplified Terms
Now, add the results from the two rationalized terms:
- Sum = \((\sqrt{2} - 1) + (\sqrt{3} - \sqrt{2})\)
- Combine like terms. The \(\sqrt{2}\) and \(-\sqrt{2}\) cancel out: \(\text{Sum} = \sqrt{3} - 1\)
Approximating the Value
To find the value closest to the options, we approximate \(\sqrt{3}\).
- We know that \(\sqrt{3} \approx 1.732\).
- Substitute this value into the simplified sum: \(\text{Value} \approx 1.732 - 1 = 0.732\)
Comparing $0.732$ to the given options, it is closest to $0.73$.
Important Questions from Surds and Indices
The value of \(\frac{{{{\left( {251} \right)}^3} + {{\left( {249} \right)}^3}}}{{25.1 \times 25.1 - 624.99 + 24.9 \times 24.9}}\) is 5 × 10 k , where the value of k is :
Find the value of m in \(\left(\frac{2}{7}\right)^{-3} \times \left(\frac{2}{7}\right)^{-5}=\left (\frac{2}{7}\right)^{-3m+1}\)
If √625 = 25; then√(.00000625/25)is:
A. 0.0025
B. 0.001
C. 0.0001
D. 0.0005Find the value of:
\(\sqrt{150}-\sqrt{54}-\sqrt{24}\)
If \(\sqrt{4624}=68\) , then the value of:
\(\sqrt{46.24}+\sqrt{0.4624}+\sqrt{0.004624}\)
