Square and square roots Grade VIII worksheet with solution

Squares and Square Roots

1. Question: Show that the following numbers is a perfect square

   (a) 2304

   (b) 3025

   Solution:

   (a) \(\sqrt{2304} = 48\), since 48 is an integer, 2304 is a perfect square.

   (b) \(\sqrt{3025} = 55\), since 55 is an integer, 3025 is a perfect square.

2. Question: Find the smallest number by which 1008 should be multiplied to make it a perfect square.

   Solution: The prime factorization of 1008 is \(2^4 \times 3^2 \times 7\). For it to be a perfect square, each prime factor must occur an even number of times. The smallest number by which 1008 should be multiplied is 7. Then, 1008 multiplied by 7 is \(2^4 \times 3^2 \times 7^2\), which is a perfect square.

3. Question: Express

   (a) \(21^2\)

   (b) \(19^2\) as the sum of two consecutive integers.

   Solution:

   (a) \(21^2 = 441 = 220 + 221\) (sum of two consecutive integers).

   (b) \(19^2 = 361 = 180 + 181\) (sum of two consecutive integers).

4. Question: Write a Pythagorean triplet whose one number is 80.

   Solution: We can create a Pythagorean triplet using the formulas \(a = 2mn\), \(b = m^2 - n^2\), and \(c = m^2 + n^2\), where \(m\) and \(n\) are positive integers with \(m > n\). For \(a = 80\), let's choose \(m = 9\) and \(n = 1\). Then, \(b = 9^2 - 1^2 = 80\) and \(c = 9^2 + 1^2 = 82\). So, one such Pythagorean triplet is \(80, 80, 82\).

5. Question: Express 324 as sum