Square root made easy

Algebra Level 2

You are given that :

441 = 21 \sqrt[]{441} = 21

Then find the value of: 470 \sqrt[]{470} approximated upto 2 decimal places

Notes : Don't use calculator or process any Long division method of square root


The answer is 21.690.

Viki Zeta by Viki Zeta , 19, India

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1 solution

Viki Zeta
Aug 13, 2016

If ’x’ is any imperfect square, y is the just-previous perfect square. Then, x ≈ 3 y − x 2 y (or) y y − x 2 y (Mixed fraction) To find 470 , we have 441 = 21 470 − 441 = 29 441 = 21 2 441 = 2 × 21 = 42 ∴ 470 ≈ 21 29 42 = ( 21 × 42 ) + 29 42 = 911 42 = 21.6904 \text{If 'x' is any imperfect square, y is the just-previous perfect square. Then, }\\ x \approx \dfrac{3y - x}{2\sqrt[]{y}} \text{ (or) } \sqrt[]{y}\dfrac{y-x}{2\sqrt[]{y}} \text{ (Mixed fraction)} \\ \text{To find } \sqrt[]{470} \text{, we have } \sqrt[]{441} = 21 \\ 470 - 441 = 29 \\ \sqrt[]{441} = 21 \\ 2\sqrt[]{441} = 2 \times 21 = 42 \\ \therefore \sqrt[]{470} \approx 21 \dfrac{29}{42} = \dfrac{(21 \times 42) + 29}{42} \\ = \dfrac{911}{42} = \fbox{21.6904}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

So, you have used Taylor's expansion essentially right?

Anand Chitrao - 4 years, 10 months ago

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Not necessarily but, yes!

Viki Zeta - 4 years, 10 months ago

Can you explain where the first formula comes from?

Calvin Lin Staff - 4 years, 10 months ago

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Yeah sure. I'll edit to soon today

Viki Zeta - 4 years, 10 months ago

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