If $$\frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {3 + \sqrt 5 } }} = a + \sqrt 5 b$$ with a, b > 0, then what is the value of (ab) : (a + b)?
A. 7 : 4
B. 7 : 8
C. 4 : 7
D. 8 : 7
Answer: Option B
Solution(By Examveda Team)
$$\eqalign{ & \frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {3 + \sqrt 5 } }} = a + \sqrt 5 b \cr & {\text{LHS}} = \frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {3 + \sqrt 5 } }} \cr & = \frac{{22\sqrt 2 }}{{4\sqrt 2 - \sqrt {\frac{{6 + 2\sqrt 5 }}{2}} }} \cr & = \frac{{22\sqrt 2 }}{{4\sqrt 2 - \frac{{\sqrt 5 + 1}}{{\sqrt 2 }}}} \cr & = \frac{{22\sqrt 2 }}{{\frac{{8 - \sqrt 5 - 1}}{{\sqrt 2 }}}} \cr & = \frac{{22 \times 2}}{{7 - \sqrt 5 }} \times \frac{{7 + \sqrt 5 }}{{7 + \sqrt 5 }} \cr & = \frac{{44\left( {7 + \sqrt 5 } \right)}}{{{7^2} - {{\left( {\sqrt 5 } \right)}^2}}} \cr & = \frac{{44\left( {7 + \sqrt 5 } \right)}}{{44}} \cr & = 7 + \sqrt 5 \cr & {\text{Compare LHS and RHS}} \cr & 7 + \sqrt 5 = a + \sqrt 5 b \cr & a = 7;\,b = 1 \cr & \left( {ab} \right):\left( {a + b} \right) = \left( {7 \times 1} \right):\left( {7 + 1} \right) = 7:8 \cr} $$Related Questions on Algebra
If $$p \times q = p + q + \frac{p}{q}{\text{,}}$$ then the value of 8 × 2 is?
A. 6
B. 10
C. 14
D. 16
A. $$1 + \frac{1}{{x + 4}}$$
B. x + 4
C. $$\frac{1}{x}$$
D. $$\frac{{x + 4}}{x}$$
A. $$\frac{{20}}{{27}}$$
B. $$\frac{{27}}{{20}}$$
C. $$\frac{6}{8}$$
D. $$\frac{8}{6}$$
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