Squares and Square Roots
ICSE Class 8 · Mathematics · Chapter 3
What this chapter covers
- Squares and perfect squares
- Properties of perfect squares
- Patterns in square numbers
- Square roots
- Square root by prime factorisation
- Square root by long division
- Square roots of decimals
- Estimating square roots
Key points
- The square of a number is the number multiplied by itself, written n2.
- A perfect square is the square of a natural number.
- A perfect square ends in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8.
- A perfect square ends in an even number of zeros.
- Between n2 and (n+1)2 lie 2n numbers that are not perfect squares.
- The sum of the first n odd numbers is n2.
- (n+1)2 - n2 = n + (n+1).
- The square root of a number is the number which, multiplied by itself, gives it; √( ) means the positive root.
- Prime factorisation: pair the equal primes and take one from each pair. An unpaired prime shows the number is not a perfect square.
- Long division: pair the digits from the units place, then double the root found so far at every step.
- For decimals, pair the whole part leftwards and the decimal part rightwards from the decimal point.
- To estimate a square root, find the perfect squares on either side and decide which is nearer.
Questions answered
Write the squares of 13 and 19.
132 = 169 and 192 = 361.
Without working it out, explain why 4327 is not a perfect square.
A perfect square never ends in 2, 3, 7 or 8. The number 4327 ends in 7, so it cannot be a perfect square.
How many natural numbers lie between 252 and 262 that are not perfect squares?
Between n2 and (n+1)2 there are 2n such numbers. With n = 25, there are 2 × 25 = 50.
Use the odd-number pattern to find 1 + 3 + 5 + dots + 29.
The odd numbers from 1 to 29 are the first 15 odd numbers. Their sum is 152 = 225.
Find 852 using the rule for numbers ending in 5.
The digit before 5 is 8, and 8 × 9 = 72. Write 25 after it: 852 = 7225.
Find √(3136) by prime factorisation.
3136 = (2 × 2) × (2 × 2) × (2 × 2) × (7 × 7). One factor from each pair gives 2 × 2 × 2 × 7 = 56.
Find the smallest number by which 1008 must be divided to make it a perfect square, and the square root of the result.
1008 = (2 × 2) × (2 × 2) × (3 × 3) × 7. The 7 has no partner, so divide by 7. The result is 144, and √(144) = 12.
Find √(1369) by long division.
The pairs are 13 and 69. The largest square in 13 is 9, so the first digit is 3, remainder 4. Bring down 69 to get 469. Double 3 to get 6; 67 × 7 = 469. So √(1369) = 37.