Square roots

Number Theory Level 2

What is the square root of 9999?

99.346 99.912 99.33 99.995 98.673

Hardik Siloiya by Hardik Siloiya , 19, India

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1 solution

We have that

999 9 1 2 = ( 10000 − 1 ) 1 2 = 1000 0 1 2 ∗ ( 1 − 1 10000 ) 1 2 = 100 ∗ ( 1 − 1 10000 ) 1 2 9999^{\frac{1}{2}} = (10000 - 1)^{\frac{1}{2}} = 10000^{\frac{1}{2}}*\left(1 - \dfrac{1}{10000}\right)^{\frac{1}{2}} = 100*\left(1 - \dfrac{1}{10000}\right)^{\frac{1}{2}} .

We can then use the approximation ( 1 − x ) n ≈ 1 − n x (1 - x)^{n} \approx 1 - nx when n x < < 1 nx \lt \lt 1 , giving us that

( 1 − 1 10000 ) 1 2 ≈ 1 − 1 2 ∗ 1 10000 = 1 − 1 20000 \left(1 - \dfrac{1}{10000}\right)^{\frac{1}{2}} \approx 1 - \dfrac{1}{2}*\dfrac{1}{10000} = 1 - \dfrac{1}{20000} . We then see that

9999 ≈ 100 ∗ ( 1 − 1 20000 ) = 100 − 1 200 = 100 − 0.005 = 99.995 \sqrt{9999} \approx 100*\left(1 - \dfrac{1}{20000}\right) = 100 - \dfrac{1}{200} = 100 - 0.005 = \boxed{99.995}

serves as the best approximation amongst the options provided.

(Using a calculator, 9999 = 99.9949999 \sqrt{9999} = 99.9949999 rounded to 7 decimal places.)

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