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Simplifying Square Roots

A square root of a number is one of two equal factors of the number. A radical sign, , is used to indicate a positive square root.

Every positive number has a positive square root and a negative square root. A negative number like 4 has no real square root because the square of a number cannot be negative.

Example:

Find the square root.

49

49 indicates the positive square root of 49 .

Since 7 × 7 is 49 , 49 = 7 .

Find the square root.

25

25 indicates the negative square root of 25 .

Since 5 × 5 is 25 , 25 = 5 .

Estimate Square Root

The square root of a perfect square is an integer. You can estimate the square root of a number that is not a perfect square .

Example:

Estimate 69 to the nearest whole number.

First list some perfect squares and look for the whole numbers near 69 .

1, 4, 9, 16, 25, 36, 49, 64, 81, 100,  ...  

Observe that 69 lies between the perfect squares 64 and 81 . That is, 64 < 69 < 81 .

Now find the square root of each number.

64 < 69 < 81 8 < 69 < 9

So, 69 is between 8 and 9 .

Since 69 is much closer to 64 than to 81 , the best whole number estimate is 8 .