Squares and Square Roots
Numbers · Squares and Square Roots
Syllabus tag: KCSE | Mathematics | Form 1 | Topic 9 Squares and Square Roots
Lesson objectives
By the end of this topic, you should be able to:
- Find the square of a number by multiplication and from tables.
- Find square roots by the factor method.
- Find squares and square roots from mathematical tables.
- Estimate square roots of numbers that are not perfect squares.
Squares and Square Roots
Squaring multiplies a number by itself. Finding a square root reverses that.
a) Squaring
To square a number, multiply it by itself. We write it with an index 2.
So 15² = 15 × 15 = 225.
For a fraction, square the numerator and the denominator. So (2/3)² = 4/9.
For a decimal, multiply and count the decimal places. So 0.4² = 0.16.
A decimal below 1 gets smaller when squared, because you are taking a part of a part.
b) Perfect squares
A perfect square comes from squaring a whole number.
The first twelve are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.
Knowing these on sight saves time in almost every later topic.
c) Square root by the factor method
Write the number as a product of primes in index form.
Halve every index. The result is the square root.
This works because squaring doubles each index, so rooting must halve them.
If any index is odd, the number is not a perfect square.
d) Square root by pairing factors
An equivalent way to see it. Write out the prime factors and split them into two identical groups.
Take 1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3. Each group is 2 × 2 × 3 × 3 = 36.
So the square root is 36.
e) Using tables
Mathematical tables give squares and square roots to four figures.
Find the row for the whole part and the column for the first decimal place. Then add the correction from the difference columns.
Tables give four figures, so results are approximate.
f) Estimating a root
Find the perfect squares immediately below and above.
The root lies between their roots.
Judge where it sits between them. Since 50 is very close to 49, √50 is a little over 7.
Estimate before using tables or a calculator. If the machine gives 70, an estimate shows at once that a key was pressed wrongly.
g) Where this is used
Pythagoras' theorem. Finding the side of a square from its area. Standard deviation later in statistics. Any formula containing a square or a root.
Words to know
- Square -- the result of multiplying a number by itself.
- Perfect square -- a number whose square root is a whole number.
- Square root -- a value which, when squared, gives the original number.
- Radical sign -- the symbol √, denoting the positive square root.
- Mean differences -- the correction columns in mathematical tables.
:::checkpoint Check yourself
- Work out 14².
- Find √576 by the factor method.
- Between which two whole numbers does √90 lie?
- Why does halving the indices give the square root? :::
Bridge to practice
The exercises begin with squares and perfect squares, move through the factor method and the two roots, and finish with tables, estimation and decimals. Before writing any square root answer, name the two perfect squares it lies between — if your answer falls outside them, it is wrong.