All questions of Squares and Square Roots for Class 6 Exam

Because if we add the number 5 five times we will get the approximate number 25

Calculating the Square Root of Two-Digit Perfect Squares
To find the square root of a perfect square number, we need to determine the number that, when multiplied by itself, gives the original number. In this case, we are looking for the square root of the two-digit perfect square numbers 16.

Process to Find the Square Root
- The square root of 16 is a number which, when multiplied by itself, gives 16.
- Let's check each option:
a) Square root of 16 = 4 x 4 = 16
b) Square root of 16 = 4 x 4 = 16
c) Square root of 16 ≠ 4.5 (as 4.5 x 4.5 = 20.25)
d) Square root of 16 ≠ 5 (as 5 x 5 = 25)

Correct Answer: Option B
Therefore, the square root of the two-digit perfect square number 16 is 4, which satisfies the condition that when multiplied by itself, gives the original number 16.

Explanation:

Duplex Method:
The duplex method is a technique used to find the square of numbers quickly and efficiently. It involves breaking down the number into two parts and performing calculations based on those parts.

Given Number: 88

Calculations:
- Step 1: Take the first part of the number, which is 88, and find the square of this part.
- 88^2 = 7744
- Step 2: Take the second part of the number, which is 8, and find the square of this part.
- 8^2 = 64
- Step 3: Multiply the first part by the second part and double the result.
- 88 * 8 * 2 = 1408
- Step 4: Combine the results from Step 1, Step 2, and Step 3 to get the final answer.
- Final Answer = 7744 | 1408 | 64 = 6944
Therefore, the square of 88 using the duplex method is 6944.

Explanation:

Duplex Method:
The duplex method is a technique used to find the square of a number quickly. It involves breaking down the number into tens and units and then calculating the square.

Given Number: 26
To find the square of 26 using the duplex method, follow these steps:

Step 1: Break down the number into tens and units.
26 = 20 + 6

Step 2: Calculate the square of the tens and units separately.
20^2 = 400
6^2 = 36

Step 3: Multiply the tens by the units and double the result.
2 * 20 * 6 = 240

Step 4: Add the results from steps 2 and 3 to get the square of the given number.
400 + 36 + 240 = 676
Therefore, the square of 26 using the duplex method is 676.

Correct Answer: 676 (Option B)

Understanding the Duplex Method
The duplex method is a systematic way to find the square of a number. It involves breaking the number into parts and using a specific formula to calculate the square.
Steps to Calculate the Square of 690
1. Break Down the Number
- Split 690 into two parts: 6 (hundreds) and 90 (tens).
2. Calculate Squares of Each Part
- Square of 6: 6 × 6 = 36
- Square of 90: 90 × 90 = 8100
3. Calculate the Cross Product
- Multiply the two parts and then double the result:
- 2 × (6 × 90) = 2 × 540 = 1080
4. Combine the Results
- Now, combine the results using the duplex method:
- The squares and the cross product will be added together:
- Square of 690 = Square of 6 + Cross product + Square of 90
- = 36 (from 6) + 1080 (cross product) + 8100 (from 90)
5. Final Calculation
- Add the values:
- 36 + 1080 + 8100 = 9180
6. Adjust for Place Values
- Since we are calculating (690)², we need to consider the place values correctly.
- The final answer will be: 690 × 690 = 476100
Conclusion
Thus, the square of 690 is 476100, which corresponds to option 'A'. This method is effective for calculating the squares of larger numbers by breaking them down into manageable parts.

Explanation:

Duplex Method:
The duplex method is a technique used to quickly find the squares of numbers ending in 5. It involves multiplying the tens digit by its successor for the first part of the answer and then appending 25 to the result to get the final answer.

Calculation:
Given number: 9448
- Tens digit: 94
- Successor of 9 (tens digit): 9+1 = 10
- First part of the answer: 94 x 10 = 940
- Append 25 to get the final answer: 94025
Therefore, the square of 9448 using the duplex method is 9448 x 9448 = 89214825.
Hence, the correct answer is option 'B'.

Explanation:

Duplex Method:

The duplex method is a technique used to find the square of numbers quickly and efficiently. It involves breaking down the number into tens and units digits, squaring them separately, and then combining the results to get the final answer.

Given number:
326

Calculations:

- Break down the number into tens and units:
- Tens digit = 32
- Units digit = 6

- Square the tens digit:
- (32)^2 = 1024

- Multiply the tens digit by the units digit and double the result:
- 2 * 32 * 6 = 384

- Square the units digit:
- (6)^2 = 36

- Combine the results:
- 1024 || 384 || 36 = 104176

Therefore, the square of 326 using the duplex method is 104176.

Solution:

To find the square root of a four-digit perfect square number, we need to determine the two-digit number whose square is equal to the given four-digit number.

Given number: 1024

Step 1: We start by finding the square root of the first two digits of the given number, which is 10. The square root of 10 is approximately 3.16.

Step 2: Next, we need to determine the correct digit for the units place. To do this, we consider the remaining two digits of the number, which is 24.

Step 3: We check the square of the number obtained in step 1 (3), by multiplying it with the next consecutive number. In this case, 3 * 4 = 12.

Step 4: We compare the value obtained in step 3 (12) with the remaining two digits of the given number (24). If the value in step 3 is less than the remaining two digits, we increment the digit of the units place by 1.

Step 5: We repeat step 3 and step 4 until the value obtained in step 3 is greater than or equal to the remaining two digits of the given number.

Step 6: From the above steps, we find that the square root of the given number 1024 is 32.

Therefore, the correct answer is option 'B' (32).

Understanding the Duplex Method:
The duplex method is a technique used to find the square of a number quickly and efficiently. It involves breaking down the number into its components and performing calculations based on these components.

Finding the Square of 976:
To find the square of 976 using the duplex method, we follow these steps:
1. Identify the base number: In this case, the base number is 970 as it is closest to 976 and is a multiple of 10.
2. Calculate the deviation: The deviation is the difference between the given number (976) and the base number (970), which is 6.
3. Find the duplex: The duplex is the square of the deviation, which is 36.
4. Calculate the cross product: Multiply the deviation by the duplex and double the result. (6 * 36) * 2 = 432.
5. Find the square: Add the duplex to the base number and add the cross product to get the square of 976. 970 + 36 = 1006, 1006 + 432 = 1438.
Therefore, the square of 976 is 946176, which corresponds to option 'A'.

Step-by-Step Explanation:

Given Number: 148

1. Decomposition:
- Write the number as 100a + 10b + c
- For 148, a = 1, b = 4, c = 8

2. Duplex Calculation:
- Add the last digit (c) to the square of the first digit (a^2) and write the result in the unit's place.
- For 148, 8 + 1^2 = 9 (Write 9 in the unit's place)
- Double the product of the first and last digit (2 * a * c) and add it to the product of the first and second digit (2 * a * b) and write the result in the ten's place.
- For 148, 2 * 1 * 8 + 2 * 1 * 4 = 16 + 8 = 24 (Write 4 in the ten's place and carry over 2)
- Add the carried-over number (if any) to the square of the middle digit (b^2) and write the result in the hundred's place.
- For 148, 2 + 4^2 = 18 (Write 8 in the hundred's place)
- If there is a carry in the hundred's place, carry it over to the thousand's place.

3. Square of the Given Number:
- Combine the results from the duplex calculation to get the square of the given number.
- For 148, the square is 21904.
Therefore, the square of 148 using the duplex method is 21904.