CBSE Class 8 Maths Exemplar Solutions Chapter 3: Square, Square Root and Cube Cube Root

To determine which number's square is 196, we can check the squares of the given options or find the square root of 196. We know that $14\; \backslash times\; 14\; =\; 196$. Therefore, 196 is the square of 14. The correct option is (C).

We need to identify which of the given numbers is the square of an even number. Let's check each option:

• $144\; =\; 12^2$. Since 12 is an even number, 144 is the square of an even number.
• $169\; =\; 13^2$. 13 is an odd number.
• $441\; =\; 21^2$. 21 is an odd number.
• $625\; =\; 25^2$. 25 is an odd number.

Therefore, the correct option is (A) because 144 is the square of the even number 12.

To find the unit digit of the square of a number ending in 9, we only need to consider the unit digit itself. We square the unit digit: $9\; \backslash times\; 9\; =\; 81$. The unit digit of 81 is 1. Thus, any number ending in 9, when squared, will have 1 as its unit digit. For example, $19^2\; =\; 361$, $29^2\; =\; 841$. The correct option is (C).

The unit digit of the square of a number is determined by the unit digit of the number itself. We examine the unit digit of each option:

• For $14^2$: The unit digit is 4. $4^2\; =\; 16$. The unit digit of $14^2$ is 6.
• For $62^2$: The unit digit is 2. $2^2\; =\; 4$. The unit digit of $62^2$ is 4.
• For $27^2$: The unit digit is 7. $7^2\; =\; 49$. The unit digit of $27^2$ is 9.
• For $35^2$: The unit digit is 5. $5^2\; =\; 25$. The unit digit of $35^2$ is 5.

Therefore, $62^2$ will have 4 at its units place. The correct option is (B).

First, we calculate the values of the squares: $5^2\; =\; 25$ and $6^2\; =\; 36$. We need to find the number of natural numbers strictly between 25 and 36. These numbers are 26, 27, 28, 29, 30, 31, 32, 33, 34, and 35. Counting these numbers, we find there are 10 natural numbers.

Alternatively, there is a formula that states the number of natural numbers between two consecutive squares $n^2$ and $(n+1)^2$ is $2n$. In this case, $n=5$, so the number of natural numbers is $2\; \backslash times\; 5\; =\; 10$. The correct option is (B).

A key property of perfect squares is that they cannot end in the digits 2, 3, 7, or 8. Let's examine the unit digits of the given numbers:

• 841 ends in 1. It is a perfect square ($29^2\; =\; 841$).
• 529 ends in 9. It is a perfect square ($23^2\; =\; 529$).
• 198 ends in 8. According to the property, a number ending in 8 cannot be a perfect square.

Therefore, 198 cannot be a perfect square. The correct option is (C).

To find the unit digit of the cube of 23, we only need to consider the unit digit of the number 23, which is 3. We then find the cube of this unit digit: $3^3\; =\; 3\; \backslash times\; 3\; \backslash times\; 3\; =\; 27$. The unit digit of 27 is 7. Therefore, the unit digit of the cube of 23 is 7. The correct option is (B).

The area of a square is calculated by the formula: Area = side × side = $side^2$. We are given that the area of the square board is 144 square units. So, we have the equation $side^2\; =\; 144$. To find the length of the side, we need to take the square root of the area: $side\; =\; \backslash sqrt\{144\}$. The square root of 144 is 12, since $12\; \backslash times\; 12\; =\; 144$. Therefore, each side of the square board is 12 units long. The correct option is (B).