Simplify: \(\sqrt{7+4\sqrt{3}}\)
\(\sqrt{3}+2\)
A nested surd of the form \(\sqrt{a+2\sqrt{b}}\) can be written as \(\sqrt{x}+\sqrt{y}\) when \(x+y=a\) and \(xy=b\).
First rewrite the inside so the surd term has a factor of 2: \(7+4\sqrt{3}=7+2\cdot 2\sqrt{3}=7+2\sqrt{12}\).
Now we need two numbers with sum \(7\) and product \(12\). These are \(4\) and \(3\), since \(4+3=7\) and \(4\times 3=12\).
Therefore \(7+4\sqrt{3}=(\sqrt{4}+\sqrt{3})^{2}=(2+\sqrt{3})^{2}\).
Taking the (positive) square root gives \(\sqrt{7+4\sqrt{3}}=2+\sqrt{3}=\sqrt{3}+2\).
Hence, the simplified value is \(\sqrt{3}+2\).
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: $(\sqrt{3} + \sqrt{2})^2$
Rationalize:
$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}$- Simplify:
$\sqrt{3}(\sqrt{12} - \sqrt{27})$ - Simplify:
$\frac{\sqrt{5+2\sqrt{6}} + \sqrt{5-2\sqrt{6}}}{\sqrt{2}}$
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:
\(\sqrt{46.24}+\sqrt{0.4624}+\sqrt{0.004624}\)
