Squaring and Other Powers

To square a number means to multiply it by itself. 3² = 3 × 3 = 9

To cube a number means to multiply it by itself by itself. 5³ = 5 × 5 × 5 = 125

See how the power tells us how many times to multiply the number by itself.

So 4 ⁵ = 4 × 4 × 4 × 4 × 4

Note: it's a good idea learn and recognize the squares of the first 12 integers
and the cubes of the first 6 integers.

(an integer is a positive or negative whole number)

.

The square of 3 is 9, but the square of –3 is also 9. That's because, any number multiplied by itself must be positive, since we're multiplying either two positives or two negatives.
If, however, we have – ( – 3)² , we will get – 9 since the negative one outside the bracket is not squared.

Square Roots

The square root of a number is the number we must square to get it.
So the square root of 9 is 3, since we must square 3 to get 9. But, as we said before,
( –3)² is also equal to 9. So it's obvious that every positive perfect square, has two square roots -- one positive, the other negative. The positive square root is called the principal square root. In this lesson and these exercises, we will use only the positive or principal square root.

The symbol we use to indicate a square root (any root actually) is like a division symbol but not quite so rounded. It looks like this:

From the table of squares, we can see that

To indicate cube root, we put a small 3 in the "v" of the symbol like this:

From the table of cubes, we can see that

Square Roots and Decimals

When we square a decimal number, the result will have twice as many decimal places as the number itself. For example, (0.2)² = 0.04, and (0.04)² = 0.0016. This therefore tells us that the square root of a decimal number has half as many decimal places as the original.
So we know that .

Examples

a)

b)

c)

d)

Estimating Square Roots

Say we need an estimate for . Since 27 lies between 25 and 36 on the number line, we know that must be between 5 and 6, the roots of 25 and 36.
So a good estimate would be 5.2 or 5.3 since 27 is much closer to 25 than it is to 36.

To estimate the value of the square root of a number that is not a perfect square, we find 2 perfect squares -- one that is less than and the other that is greater than the number in question.
Say we need to estimate the value of . We know it will be between 11 and 12 since
11² = 121 and 12² = 144. Again, since 129 is much closer to 121 than it is to 144, we would estimate that the number we need is about 11.3 or 11.4. When we check with our calculators, we find that the square root of 129 = 11.35782 which would be rounded to 11.4 if we were rounding to the nearest tenth.

Examples

Estimate, then find the actual value with a calculator. Round to the nearest tenth:

a) Since 73 is between 64 and 81, its root is between 8 and 9: estimate 8.4
with the calculator, we find which rounds to 8.5.

b) Since 411 is between 400 and 441, its root is between 20 and 21: estimate 20.3
with the calculator, we find which rounds to 20.3

c) Since 0.5 is very close to 0.49, its root is close to 0.7: estimate 0.71
with the calculator, we find which rounds to 0.7

Practice

1) Without using a calculator, find:

a)

b)

c)

d)

2) Find a number with a square root between: (there are lots of different answers)

a) 7 and 8

b) 9 and 10

c) 3 and 4

d) 5 and 6

3) Write the two whole numbers (integers) between which these roots are found:

a)

b)

c)

d)

4) A square has an area of 121 cm²
a) How long is each side?
b) What is the radius of the largest circle that will fit inside it?

Solutions

1) Without using a calculator, find:

a)

b)

c)

d)

2) Find a number with a square root between:

a) any # between 49 and 64

b) any # between 81 and 100

c) any # between 9 and 16

d) any # between 25 and 36

3)

a) 7 and 8

b) 5 and 6

c) 6 and 7

d) 10 and 11

4) a) each side of the square is 11 cm.
b) the radius is ½ of 11 or 5.5 cm.

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