Simplify: \(\sqrt{12 + 6\sqrt{3}}\)
\(3 + \sqrt{3}\)
Assume \(\sqrt{12 + 6\sqrt{3}} = a + b\sqrt{3}\) for rational a, b.
Squaring: \((a + b\sqrt{3})^2 = a^2 + 3b^2 + 2ab\sqrt{3}\)
Equating with \(12 + 6\sqrt{3}\): \(a^2 + 3b^2 = 12\) and \(2ab = 6 \Rightarrow ab = 3\).
Try \(a = 3,\ b = 1\): \(9 + 3 = 12\) ✓ and \(3 \times 1 = 3\) ✓.
Hence \(\sqrt{12 + 6\sqrt{3}} = 3 + \sqrt{3}\).
Similar Questions
If \(x = \sqrt{\dfrac{2 + \sqrt{3}}{2 - \sqrt{3}}}\), determine the value of \(x^2 + x - 9\).
Evaluate: \(\sqrt{12 + \sqrt{12 + \sqrt{12 + \cdots}}}\)
- If $x = 2 + \sqrt{3}$, find $x - \frac{1}{x}$.
- The value of $\sqrt{11 - 2\sqrt{30}} - \frac{1}{\sqrt{11 - 2\sqrt{30}}}$ is:
- What will come in place of ? to satisfy the equation : $(\sqrt{6}-1)^2 = ? - 2\sqrt{6}$
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$\frac{\sqrt{5+2\sqrt{6}}}{\sqrt{5-2\sqrt{6}}}$ Find the value of \(\sqrt{388 + \sqrt{127 + \sqrt{289}}}\).
Simplify the expression: \(\sqrt{9 + 4\sqrt{5}}\)
Choose the correct relation:
(i) \(\sqrt{10} < 3.2\)
(ii) \(\sqrt{11} < \sqrt{12}\)
(iii) \(\sqrt{13} < \sqrt{14}\)
Simplify:
\(\dfrac{1}{\sqrt{7}-\sqrt{3}} + \dfrac{1}{\sqrt{7}+\sqrt{3}}\)
Important Questions from Surds and Indices
The cube root of - 64 × - 1331 is:
If (27) m = (81) n, then m 2: mn = ?
if 49 n + 49 n + 49 n + 49 n + 49 n + 49 n + 49 n = 7 2221 , then n = ?
If a x= b y= c zand b 2= ac then y =
The value of (0.3) [{(200 - 146)/(3 × 3 × 3)} - 3] is:
