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

Answer: (d)

x = $√{72+√{72+√{72 +...}}}$

On squaring both sides,

$x^2 = 72+√{72+√{72+√{72 +...}}}$

$x^2$ = 72 + x

$x^2$ - x - 72 = 0

$x^2$ - 9x + 8x - 72 = 0

x (x - 9) + 8 (x - 9) = 0

(x + 8) (x - 9) = 0

x = 9 because x ≠ - 8

Using Rule 25

$√{72+√{72+√{72 +...}}}$ = 9

It is because 72 = 8×9 = n (n + 1)

Answer: (c)

m = $√{5+√{5+√{5 +...}}}$

On squaring both sides,

$m^2 = 5 + m ⇒ m^2$ - m = 5 ....(i)

Again,

n = $√{5-√{5-√{5-...}}}$

On squaring both sides,

$n^2$ = 5 - n

$n^2$ + n = 5 .........(ii)

$m^2$ - m = $n^2$ + n

$(m^2 - n^2)$ = m + n

(m + n) (m - n) - (m + n) = 0

(m + n) (m - n - 1) = 0

Answer: (b)

Let x = $√{3√{3√{3...}}}$

Squaring both sides,

$x^2 = 3√{3√{3√{3...}}}$ = 3x

$x^2$ - 3x = 0

x (x - 3) = 0

x = 3 because x ≠ 0

$√{3+√{3+√{3+...∞}}}$ = 3

It is because, here

n = ∞ and x =3

$√{3+√{3+√{3+...∞}}}=3^(1-1/{3∞})$

= $3^(1 - 0)$ [${something}/∞ = 0$] = 3

Answer: (c)

Expression

=$√{10+√{ 25+√{ 108+√{ 154+√{ 225}}}}}$

=$√{10+√{ 25+√{ 108+√{ 154+15}}}}$

=$√{10+√{ 25+√{ 108+√{169}}}}$

=$√{10+√{ 25+√{ 108+13}}}$

=$√{10+√{ 25+√{121}}}$

=$√{10+√{ 25+11}}$

=$√{10+6}=√16=4$

Answer: (d)

Let x = $√{6+√{6+√{6 +...}}}$

Squaring on both sides,

$x^2 = 6+√{6+√{6+√{6 +...}}}$

$x^2$ = 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

$√{6+√{6+√{6 +...}}}$ = 3

It is because

6 = 2 × 3 = n (n + 1)