model 5 simplifying roots of roots Practice Questions Answers Test with Solutions & More Shortcuts

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: (a)

Let x = $√{30+√{30+√{30+...}}}$

On squaring both sides,

$x^2 = 30+√{30+√{30+√{30+...}}}$

$x^2$ = 30 + x

$x^2$ - x - 30 = 0

$x^2$ - 6x + 5x - 30 = 0

x (x - 6) + 5 (x - 6) = 0

(x - 6) (x + 5) = 0

x = 6 because x ≠ - 5

Using Rule 25

$√{30+√{30+√{30+...}}}$ = 6

It is because

30 = 5 × 6 = n (n +1)

Answer: (b)

Let, x = $√{6+√{6+√{6 +...∞}}}$

On squaring,

$x^2 = 6 + √{6+√{6+√{6 +...∞}}}$

$x^2$ = 6 + x

$x^2$ - x - 6 = 0

$x^2$ - 3x + 2x - 6 = 0

x (x - 3) + 2 (x - 3) = 0

(x + 2) (x - 3) = 0

x = 3 because x ≠ - 2

Using Rule 25

$√{6+√{6+√{6 +...}}}$ = 3

It is because, 6 = 2 × 3 =n (n+1)