11 ⋯ 11 − 2 ⋯ 2 = ? \sqrt{11\cdots11-2\cdots 2} = ?

Algebra Level 2

1111111111111111 ⏟ Number of 1=16 − 22222222 ⏟ Number of 2=8 = ? \large \sqrt{\underbrace{1111111111111111}_{\text{Number of 1=16}} - \underbrace{22222222}_{\text{Number of 2=8}}} =\ ?

Solve it without using a calculator?

77777777 17171717 13131313 33333333

Samuel Nascimento by Samuel Nascimento , 24, Brazil

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2 solutions

Chew-Seong Cheong
Jul 21, 2019

X = 111 ⋯ 111 ⏟ No. of 1=16 − 222 ⋯ 222 ⏟ No. of 2=8 Let a = 111 ⋯ 111 ⏟ No. of 1=8 = 1 0 8 a + a − 2 a = 1 0 8 a − a = ( 1 0 8 − 1 ) a = 999 ⋯ 999 a ⏟ No. of 9=8 = 9 a 2 = 3 a = 33333333 \begin{aligned} X & = \sqrt{\underbrace{111\cdots111}_{\text{No. of 1=16}} - \underbrace{222\cdots222}_{\text{No. of 2=8}}} & \small \color{#3D99F6} \text{Let }a = \underbrace{111\cdots111}_{\text{No. of 1=8}} \\ & = \sqrt{10^8a+a-2a} = \sqrt{10^8a-a} \\ & = \sqrt{(10^8-1)a} = \underbrace{\sqrt{999\cdots999}a}_{\text{No. of 9=8}} \\ & = \sqrt {9a^2} = 3a = \boxed{33333333} \end{aligned}

2 Helpful 0 Interesting 2 Brilliant 0 Confused
Vin Benzin
Jul 20, 2019

3 Helpful 1 Interesting 1 Brilliant 0 Confused

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