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# Estimating Square Roots

Alignments to Content Standards: 8.NS.A

Without using the square root button on your calculator, estimate $\sqrt{800}$ as accurately as possible to $2$ decimal places.

(Hint: It is worth noting that $20^2 = 400$ and $30^2=900$.)

## IM Commentary

By definition, the square root of a number $n$ is the number you square to get $n$. The purpose of this task is to have students use the meaning of a square root to find a decimal approximation of a square root of a non-square integer. Students may need guidance in thinking about how to approach the task.

## Solutions

Solution: Using the definition of a square root

We know that $$20^2=400$$ and $$30^2=900$$ so $$20 \lt \sqrt{800} \lt 30$$

Choosing successive approximations carefully, we see that:

$n$$n^2 m^2 m 2878485129 28.2795.24800.8928.3 28.28799.7584800.324128.29 28.284799.984656800.04122528.285 So \sqrt{800} \approx 28.28. Solution: Another approach We know that 20^2=400 and 30^2=900, so$$20 \lt \sqrt{800} \lt 30.$$If we take the average of 20 and 30, we get \frac{20+30}{2} = 25. Since 25^2 = 625, we know that$$25 \lt \sqrt{800} \lt 30.$$If we take the average of 25 and 30, we get \frac{25+30}{2} = 27.5. Since 27.5^2 = 756.25, we know that$$27.5 \lt \sqrt{800} \lt 30.$$If we take the average of 27.5 and 30, we get \frac{27.5+30}{2} = 28.75. Since 28.75^2 = 826.5625, we know that$$27.5 \lt \sqrt{800} \lt 28.75.$$Continuing in this way, we get$$\sqrt{800} \approx 28.28.$\$

#### yuchun says:

over 4 years

we can also use more quickly method to get the results, if you don't mind, I can post my solution here..

#### Cam says:

Of course, you're more than welcome to post your solution here.

#### hollandr says:

over 6 years

Just an editing comment. In the second level of the approximations table, the last column, m, should read 28.3, not 29.3.

over 6 years

Fixed. Thanks!