Calculate $\frac{1}{x}$
Given the value of $x$ is $2 + \sqrt{3}$. To find $\frac{1}{x}$, we calculate the reciprocal:
$ \frac{1}{x} = \frac{1}{2 + \sqrt{3}} $To simplify this fraction, we multiply the numerator and the denominator by the conjugate of the denominator, which is $2 - \sqrt{3}$:
$ \frac{1}{x} = \frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} $Using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$ for the denominator:
$ \frac{1}{x} = \frac{2 - \sqrt{3}}{2^2 - (\sqrt{3})^2} = \frac{2 - \sqrt{3}}{4 - 3} = \frac{2 - \sqrt{3}}{1} $Therefore, $\frac{1}{x} = 2 - \sqrt{3}$.
Calculate $x - \frac{1}{x}$
Now, substitute the values of $x$ and $\frac{1}{x}$ into the expression $x - \frac{1}{x}$:
$ x - \frac{1}{x} = (2 + \sqrt{3}) - (2 - \sqrt{3}) $Distribute the negative sign:
$ x - \frac{1}{x} = 2 + \sqrt{3} - 2 + \sqrt{3} $Combine like terms:
$ x - \frac{1}{x} = (2 - 2) + (\sqrt{3} + \sqrt{3}) $ $ x - \frac{1}{x} = 0 + 2\sqrt{3} $Thus, the final value is $2\sqrt{3}$.
Similar Questions
- 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: $(\sqrt{3} + \sqrt{2})^2$
Rationalize:
$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}$- 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:
\(\sqrt{46.24}+\sqrt{0.4624}+\sqrt{0.004624}\)
