Simplify the expression: \(\sqrt{9 + 4\sqrt{5}}\)
\(\sqrt{5} + 2\)
Step 1 – express the radicand as a perfect square:
Try \((a + b)^2 = a^2 + b^2 + 2ab = 9 + 4\sqrt{5}\).
This needs \(a^2 + b^2 = 9\) and \(2ab = 4\sqrt{5}\).
Step 2 – pick a, b:
Take \(a = 2,\ b = \sqrt{5}\): then \(a^2 + b^2 = 4 + 5 = 9\) ✓ and \(2ab = 2 \cdot 2 \cdot \sqrt{5} = 4\sqrt{5}\) ✓.
Step 3 – take the square root:
\(\sqrt{9 + 4\sqrt{5}} = \sqrt{(2 + \sqrt{5})^2} = 2 + \sqrt{5} = \sqrt{5} + 2\).
Hence the answer is \(\sqrt{5} + 2\).
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 : $(\sqrt{6}-1)^2 = ? - 2\sqrt{6}$
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$\frac{\sqrt{5+2\sqrt{6}}}{\sqrt{5-2\sqrt{6}}}$ - 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:
\(\sqrt{46.24}+\sqrt{0.4624}+\sqrt{0.004624}\)
