Square and square roots Grade VIII Worksheet with solutions Part-2

Solutions

1. Question: Express 324 as the sum of 18 odd numbers.

   Solution: Let's express 324 as the sum of 18 odd numbers. Let the first odd number be \(a\), and since odd numbers have a common difference of 2, the rest of the numbers will be \(a + 2\), \(a + 4\), ..., \(a + 34\). Sum of these 18 odd numbers is 324:

   \[ 18a + 2(1 + 2 + 3 + \ldots + 17) = 324 \] \[ 18a + 2 \times \frac{17 \times 18}{2} = 324 \] \[ 18a + 306 = 324 \] \[ 18a = 18 \] \[ a = 1 \] So, the 18 odd numbers are 1, 3, 5, ..., 35.

2. Question: Without calculating, find the number of digits in the square root of 368645.

   Solution: We can write 368645 as \(3.68645 \times 10^5\). Now, the square root of 368645 will be the square root of this expression:

   \[ \sqrt{3.68645 \times 10^5} = \sqrt{3.68645} \times \sqrt{10^5} = \sqrt{3.68645} \times 10^{\frac{5}{2}} = \sqrt{3.68645} \times 10^{2.5} \] As \(10^{2.5}\) is between \(10^2\) and \(10^3\), the square root will have a value between 100 and 1000, which means it will have 3 digits before the decimal point.