Rationalize:
$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}$
$-5 - 2\sqrt{6}$
Rationalize Denominator of $\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}$
To rationalize the denominator $\sqrt{2}-\sqrt{3}$, we multiply both the numerator and the denominator by its conjugate, which is $\sqrt{2}+\sqrt{3}$.
The expression is:
$ \frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}} $Step 1: Multiply by Conjugate
Multiply the numerator and denominator by the conjugate of the denominator ($\sqrt{2}+\sqrt{3}$):
$ \frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}} \times \frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}+\sqrt{3}} $Step 2: Simplify Numerator
Expand the numerator:
$ (\sqrt{2}+\sqrt{3})(\sqrt{2}+\sqrt{3}) = (\sqrt{2})^2 + 2(\sqrt{2})(\sqrt{3}) + (\sqrt{3})^2 $ $ = 2 + 2\sqrt{6} + 3 $ $ = 5 + 2\sqrt{6} $Step 3: Simplify Denominator
Expand the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$:
$ (\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2 $ $ = 2 - 3 $ $ = -1 $Step 4: Combine Numerator and Denominator
Combine the simplified numerator and denominator:
$ \frac{5 + 2\sqrt{6}}{-1} $Step 5: Final Result
Simplify the final expression:
$ -(5 + 2\sqrt{6}) = -5 - 2\sqrt{6} $The rationalized form of the expression is $-5 - 2\sqrt{6}$.
Similar Questions
- If $x = 2 + \sqrt{3}$, find $x - \frac{1}{x}$.
- Simplify:
$\sqrt{12} + \sqrt{75}$ - What will come in place of ? to satisfy the equation : $(\sqrt{6}-1)^2 = ? - 2\sqrt{6}$
- Simplify:
$\frac{\sqrt{5+2\sqrt{6}}}{\sqrt{5-2\sqrt{6}}}$ - The value of $\sqrt{11 - 2\sqrt{30}} - \frac{1}{\sqrt{11 - 2\sqrt{30}}}$ is:
- If $x = \sqrt{2} + 1$, find $x^2 - 2$.
- Simplify: $(\sqrt{3} + \sqrt{2})^2$
- Simplify:
$\sqrt{3}(\sqrt{12} - \sqrt{27})$ - Simplify:
$\frac{\sqrt{5+2\sqrt{6}} + \sqrt{5-2\sqrt{6}}}{\sqrt{2}}$ Simplify: \(\sqrt{7+4\sqrt{3}}\)
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:
\(\sqrt{46.24}+\sqrt{0.4624}+\sqrt{0.004624}\)
