How do you estimate the square root of a number that is not a perfect square?

Step-by-step explanation:

Many mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring, which we learned about in a previous lesson (exponents), has an inverse too, called "finding the square root." Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …

The square root of a number, n, written

√n the number that gives n when multiplied by itself. For example,

√100=10 because 10 x 10 = 100

Examples

Here are the square roots of all the perfect squares from 1 to 100.

√ 1 = 1 since 1 ² = 1

√ 4 = 2 since 2 ² = 4

√ 9= 3 since 3 ² = 9

√ 16= 4 since 4 ² = 16

√ 25= 5 since 5 ² = 25

√ 36= 6 since 6 ² = 36

√ 49= 7 since 7 ² =29

√ 64= 8 since 8 ² = 64

√ 81= 9 since 9 ² = 81

√ 100= 10 since 10 ² = 100

Finding square roots of of numbers that aren't perfect squares without a calculator

1. Estimate - first, get as close as you can by finding two perfect square roots your number is between.

2. Divide - divide your number by one of those square roots.

3. Average - take the average of the result of step 2 and the root.

4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.

Example: Calculate the square root of 10 () to 2 decimal places.

1. Find the two perfect square numbers it lies between.

Solution:

32 = 9 and 42 = 16, so lies between 3 and 4.

2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer)

3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667

Repeat step 2: 10/3.1667 = 3.1579

Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623

Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001

If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.