model 5 simplifying roots of roots Section-Wise Topic Notes With Detailed Explanation And Example Questions

Answer: (a)

Let x = $√{2+√{2+√{2 +...}}}$

On squaring both sides

$x^2 = 2+√{2+√{2+√{2 +...}}}$

$x^2$ = 2 + x

$x^2$ - x - 2 = 0

$x^2$ - 2x + x - 2 = 0

x (x - 2) + 1 (x - 2) = 0

(x - 2) (x + 1) = 0

x = 2 or - 1

But sum of positive numbers can't be negative.

x = 2

Answer: (b)

Expression

=$√{9 + 2√{16} +√^3{ 512}}$

=$√{9 + 2√{4×4} +√^3{8×8×8}}$

=$√{9 + 2×4+8}$

=$√25$ =5

Answer: (d)

Expression

=$√{ - √{3}+√{ 3+8√{7 +4√{3}}}}$

=$√{ - √{3}+√{ 3+8√{4+3+2×2×√{3}}}}$

=$√{ - √{3}+√{ 3+8√{(2)^2+(√3)^2+2×2×√{3}}}}$

=$√{ - √{3}+√{ 3+8√{(2+√3)^2}}}$

=$√{ - √{3}+√{ 3+16+8√3}}$

=$√{ - √{3}+√{(4)^2+(√3)^2+2×4×√3}}$

=$√{ - √{3}+√{(4+√3)^2}}$

=$√{ - √{3}+4+√3}=√4=2$

Answer: (b)

${√{10+√{ 25+√{ 108+√{ 154+√{ 225}}}}}}/√^3{8}$= ?

?=${√{10+√{ 25+√{ 108+√{ 154+15}}}}}/√^3{2×2×2}$

=${√{10+√{ 25+√{ 108+√{169}}}}}/2$

=${√{10+√{ 25+√{ 108+13}}}}/2$

=${√{10+√{ 25+√{121}}}}/2$

=${√{10+√{ 25+11}}}/2$

=${√{10+√{36}}}/2={√{10+6}}/2$

=${√{16}}/2=4/2=2$

Answer: (c)

x = $√{12+√{12+√{12 +...}}}$

On squaring both sides,

$x^2 = 12 + √{12+√{12+√{12 +...}}}$

$x^2$ = 12 + x

$x^2$ - x - 12 = 0

$x^2$ - 4x + 3x - 12 = 0

x (x - 4) + 3 (x - 4) = 0

(x - 4) (x + 3) = 0

x = 4 because x ≠ - 3

Answer: (b)

Let x = $√{12+√{12+√{12 +...}}}$

On squaring both sides,

$x^2 =12+√{12+√{12+√{12 +...}}}$

$x^2$ = 12 + x

$x^2$ - x - 12 = 0

$x^2$ - 4x + 3x - 12 = 0

x (x - 4) + 3 (x - 4) = 0

(x - 4) (x + 3) = 0

x = 4, - 3

The given expression is positive.

x = 4

$√{12+√{12+√{12}}}$=4

It is because

12 = 3 × 4 = n (n + 1)