Squares and Square Roots

Syllabus tag: Kenya CBC | Grade 7 Mathematics | Strand 1.0 Numbers | Sub-Strand 1.5 Squares and Square Roots (5 lessons)

Lesson objectives

By the end of this topic, you will be able to:

• Determine the squares of whole numbers, fractions and decimals by multiplication
• Determine the square roots of whole numbers, fractions and decimals that are perfect squares
• Appreciate the use of squares and square roots in real life

Squares and Square Roots

Squaring a number means multiplying it by itself. Finding a square root reverses that.

a) Squaring whole numbers

Write it with a small 2 above and to the right. Say 9² as "nine squared".

Note that 9² is 81, not 18. Squaring is not doubling.

The first ten square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Learning them saves time everywhere.

b) Squaring fractions

Square the top and square the bottom separately.

So (2/3)² is 4/9.

For a mixed number, change it to an improper fraction first. So (1½)² becomes (3/2)², which is 9/4.

c) Squaring decimals

Multiply the decimal by itself and count the decimal places as usual.

So 0.3² is 0.09, because 3 × 3 = 9 and there are two decimal places in total.

A decimal smaller than 1 gets smaller when squared. 0.3 squared is 0.09, which is less than 0.3.

d) Square roots

The square root of a number is the value that was squared to make it. The symbol is √.

So √64 = 8, because 8 × 8 = 64.

e) Roots of fractions and decimals

For a fraction, take the root of the top and the root of the bottom.

So √(4/25) is 2/5, because √4 = 2 and √25 = 5.

For a decimal, look for a decimal that squares to give it. √0.09 = 0.3, since 0.3 × 0.3 = 0.09.

A useful check: the number of decimal places in the root is half the number in the original. 0.09 has two, so its root has one.

f) Perfect squares only

At this level every square root asked for is exact. The number under the root is a perfect square, or a fraction or decimal that works out neatly.

If your answer is not exact, check the question. You have probably misread a digit.

g) Where this is used

A tiler finds the side of a square floor from its area. Anyone using Pythagoras squares and roots numbers. Areas of squares appear throughout measurement.

Words to know

• Square (of a number) — the result of multiplying a number by itself.
• Square root — a number which, when multiplied by itself, gives the original number.
• Perfect square — a number that is the square of a whole number.
• Factor method — finding a square root by pairing identical prime factors.
• Division method — finding a square root by working through digits systematically, similar to long division.

:::checkpoint Check yourself

1. Work out 12².
2. Work out (3/4)².
3. Find √121.
4. Find √(9/16). :::

Bridge to practice

Try the exercises below — on squaring numbers and finding square roots of perfect squares.