Squares and Square Roots - Mathematics Form 1 Notes

• Squares
  • Square from Tables
• Square Roots
  • Square Roots by Factorization.
  • Square Root from Tables
• Past KCSE Questions on the Topic

Squares

• The square of a number is simply the umber multiplied by itself once. For example the square of 1 5 is 225.That is 15 x 15 = 225.

Square from Tables

• The squares of numbers can be read directly from table of squares. This tables give only approximate values of the squares to 4 figures. The squares of numbers from 1 .000 to 9.999 can be read directly from the tables.
• The use of tables is illustrated below

Example

Find the square of:

1. 4.25
2. 42.5
3. 0.425

Tables

1. To read the square of 4.25, look for 4.2 down the column headed x. Move to the right along this row, up to where it intersects with the column headed 5.The number in this position is the square of 4.25
   So 4.25² = 18.06 to 4 figures
2. The square of 4.25 lies between 40² and 50² between 1600 and 2500.
   42.5²= (4.25 x 10¹)²
   = 4.25² x 10²
   = 18.06 x 100
   = 1806
3. 0.425 2= (4.25 x ¹/₁₀)²
   = 4.25² x(¹/₁₀ )²
   =18.06 x ¹/₁₀₀
   = 0.1806

The square tables have extra columns labeled 1 to 9 to the right of the thick line.

The numbers under these columns are called mean differences.

To find 3.162, read 3.1 6 to get 9.986.Then read the number in the position where the row containing 9.986 intersects with the differences column headed 2. The difference is 13 and this should be added to the last digits of 9.986

9.986
+  13
9.999

56.129 has 5 significant figures and in order to use 4 figures tables, we must first round it off to four figures.
56.129 = 56.13 to 4 figures
56.13² = (5.613 x 10¹) ²
= 31.50 x 10²
= 3150

Square Roots

• Square roots are the opposite of squares. For example 5 x 5 = 25, we say that 5 is a square root of 25.
• Any positive number has two square roots, one positive and the other negative .The symbol for the square root of a number is √.
• A number whose square root is an integer is called a perfect square. For example 1, 4, 9, 25 and 36 are perfect squares.

Square Roots by Factorization.

• The square root of a number can also be obtained using factorization method.

Example

Find the square root of 81 by factorization method.

Solution

√81 = 3 x 3 x 3 x3 (Find the prime factor of 81 )
= (3x3) (3 x 3) (Group the prime factors into two identical numbers)
= 3 x 3 (Out of the two identical prime factors, choose one and find their product)
= 9

Note

• Pair the prime factors into two identical numbers. For every pair, pick only one number then obtain the product.

Example

Find √1764 by factorization.

Solution

1764 = 2 x 2 x 3 x 3 x 7 x 7
= 2 x 3 x 7
= 42

Example

Find √441 by factorization

Solution

√441 = 3 x 3 x 7 x 7
= 3 x 7
=21

Square Root from Tables

• Square roots of numbers from 1.0 to 99.99 are given in the tables and can be read directly.

Examples

Use tables to find the square root of:

1. 1.86
2. 42.57
3. 359
4. 0.8236

Solution

1. To read the square root of 1 .86, look for 1 .8 in the column headed x, move to the right along this row to where it intersects with the column headed 6.The number in this position is the square root of 1.86.Thus 1.86 = 1.364 to 4 figures.
2. 42.57Look for 42 in the column headed x and move along the row containing 42 to where it intersects with the column headed 5.Read the number in this position, which is 6.519. The difference for 7 from the difference column along this row is 6. The difference is added to 6.519 as shown below:
      6.519
   + 0.006
       6.525
   Thus, √42.57 = 6.525 to 4 figures.
   For any number outside this range, it is necessary to first express it as the product of a number in this range and an even power of 10.
3. 359 = 3.59 x 10²
   √359 = √(3.59 x 100)
   = 1.895 x 10
   = 18.95 (four figures)
4. 0.8236 = 82.36 x (¹/₁₀)²
   √0.8236 =√(82.36 x ¹/₁₀₀)
   = (9.072 + 0.004) x ¹/₁₀
   = 0.9076 (4 figures)

Past KCSE Questions on the Topic

1. Evaluate without using tables or calculators
   
   
2. Evaluate using reciprocals, square and square root tables only.
   
   
3. Using a calculator, evaluate (Show your working at each stage)
   
   
4. Use tables of reciprocals and square roots to evaluate
   
   
5. Use tables to find;
   1. 1. 4.978²
      2. The reciprocal of 31 .65
   2. Hence evaluate to 4.S.F the value of
      4.978² – ¹/_31.65
6. Use tables of squares, square roots and reciprocals to evaluate correct to 4 S.
         3     −   2   
   √0.0136   3.72²
7. Without using mathematical tables or calculator, evaluate:
   giving your answer in standard form