CBSE Class 8 Mathematics Square and Square Roots Notes Set 13

Squares and Perfect Squares

A natural number is a perfect square when it equals some natural number multiplied by itself, for example 49 = 7 × 7. The square root reverses the squaring: the square root of 49 is 7. In this class we use only the positive root.
Unit-digit test: a perfect square can end only in 0, 1, 4, 5, 6 or 9. A number ending in 2, 3, 7 or 8 is never a square, and a number ending in an odd number of zeros is never a square.
The unit digit of a square depends only on the unit digit of the number: bases ending in 1 or 9 give 1, in 2 or 8 give 4, in 3 or 7 give 9, in 4 or 6 give 6, in 5 give 5 and in 0 give 0. Squares of even numbers are even and squares of odd numbers are odd.

Patterns in Square Numbers

Sum of odd numbers: the sum of the first n odd numbers is n², so 1 + 3 + 5 + 7 + 9 = 25 = 5². A number that cannot be built by subtracting 1, 3, 5, ... in turn down to exactly zero is not a perfect square.
Gaps between squares: there are exactly 2n whole numbers between n² and (n + 1)², and all of them are non-squares.
Squares of odd numbers: n² = (n² − 1)/2 + (n² + 1)/2, so 7² = 49 = 24 + 25 and 11² = 121 = 60 + 61.
Squares ending in 5: the square of a5 is a(a + 1) followed by 25, so 35² = 3 × 4 | 25 = 1225.
Triangular numbers: two consecutive triangular numbers add up to a square, for example 3 + 6 = 9 and 6 + 10 = 16.
Products of numbers on either side of a middle number: (a − 1)(a + 1) = a² − 1, so 11 × 13 = 12² − 1 = 143.
Pythagorean triplets: for any natural number m greater than 1, the numbers 2m, m² − 1 and m² + 1 form a triplet, because (2m)² + (m² − 1)² = (m² + 1)².

Square Root by Prime Factorisation

Write the number as a product of primes, pair equal primes, take one prime from each pair and multiply. If a prime is left unpaired, the number is not a perfect square. To make it one, multiply by the unpaired primes; or divide by them to get a smaller square.

Example 1. Find the smallest number by which 4050 must be multiplied to make it a perfect square, and find the root of the new number.
Answer:
4050 = 2 × 3⁴ × 5². Only the prime 2 is unpaired.
Multiply by 2: 8100 = 2² × 3⁴ × 5², and the root is 2 × 9 × 5 = 90.

Example 2. Find the smallest number by which 9408 must be divided to make it a perfect square, and find the root of the quotient.
Answer:
9408 = 2⁶ × 3 × 7². Only 3 is unpaired.
9408 ÷ 3 = 3136 = 2⁶ × 7², and the root is 8 × 7 = 56.

Example 3. Find the smallest square number divisible by each of 6, 9 and 15.
Answer:
LCM = 90 = 2 × 3² × 5. The primes 2 and 5 are unpaired, so multiply by 10.
90 × 10 = 900 = 30².

Square Root by Long Division

1. Mark the digits in pairs starting from the units place.
2. Find the largest digit whose square is not more than the first pair. Write it as the first digit of the root and subtract its square.
3. Bring down the next pair beside the remainder. Double the root so far and write it as the start of the new divisor.
4. Choose the largest digit that, written next to the divisor and multiplied by itself as the new digit, does not exceed the dividend. Repeat until all pairs are used.
Number of digits: a perfect square with n digits has a root with n/2 digits if n is even and (n + 1)/2 digits if n is odd.
Decimals: pair the digits on both sides of the decimal point, so 7.29 = 729/100 has root 27/10 = 2.7.

Example 4. Find the square root of 5476 by long division.
Answer:
Pairs: 54 | 76. The largest square not exceeding 54 is 49 (7²); remainder 5.
Bring down 76 to get 576. Double 7 to get 14; 144 × 4 = 576, so the next digit is 4.
The root is 74.

Example 5. What must be added to 7581 to make it a perfect square?
Answer:
87² = 7569 < 7581 < 88² = 7744.
Least number to be added = 7744 − 7581 = 163.

Example 6. Find the greatest 4-digit perfect square.
Answer:
99² = 9801 and 100² = 10000 has five digits, so the answer is 9801.