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Square roots (estimating)

Finding the square root of a perfect square is easy.  What number, times itself, equals your number?

$$\eqalign{\sqrt{81}&=9\\\sqrt{400}&=20}$$

But, what do you do when you need to find the square root of a non-perfect square.

If you have a calculator, it's simple, you use the $\sqrt{\text{  }}$ button.

If you don't have a calculator, you need to estimate.  You won't get a perfect answer by estimating but you can come close. 

To estimate a non-perfect square root, first find the perfect squares that are closest to your number (the one that is closest but bigger than your number, and the one that is closest and smaller than your number).

 

So, let's say you want to estimate the square root of 70.

$\sqrt{70}\approxeq \text{?}$

 

What are the perfect squares that are closest to 70?

The nearest perfect square that is greater than 70 is 81.

The nearest perfect square that is less than 80 is 64.

$\sqrt{81}=9$

$\sqrt{64}=8$


 

So we know that the square root of 70 is between 8 and 9.  It's 8.?

We can refine this a bit more.  If 70 is right in the middle between 8 81 and 64, then the square root is about 8.5.   If it's closer to 81, then the decimal is higher. If it's closer to 64, then the decimal is lower.  

 

$70 - 6 = 64$

$70 + 11 = 81$

 

70 is almost twice as far from 81 as it is from 64, so it's definitely closer to 64.  Let's estimate the squart root as:

 

$\sqrt{70}\approxeq 8.25$


 

Now, let's use a calculator and see how close we were:


 

$\sqrt{70}=8.3666$


 

Pretty close.


 

Next time you want to estimate a square root, just find the nearest perfect squares and know that your square root is between their perfect square roots.  Then judge the distance and pick a decimal place.


 

Practice Problems:

  • Square roots (estimating)

    Between which two integers does the square root lie?

    1. $\sqrt{6}$

    2.  $\sqrt{27}$

    3. $\sqrt{95}$

    4. $\sqrt{31}$

    5. $\sqrt{460}$

    6. $\sqrt{291}$

    7. $\sqrt{115}$

    8.$\sqrt{1021}$

    9. $\sqrt{195}$

    10 $\sqrt{152}$

    Find or approximate the square root.  Estimate to the nearest tenth.

    11. $\sqrt{576}$

    12. $\sqrt{169}$

    13. $\sqrt{71}$

    14. $\sqrt{50}$

    15. $\sqrt{441}$

    16. $\sqrt{961}$

    17. $\sqrt{123}$

    18. $\sqrt{135}$

    19. $\sqrt{225}$

    20. $\sqrt{365}$

Skill:

Common Core Grade Level/Subject

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