Using a shortcut method, find the squares of the following numbers.A) 35B) 65C) 145D) 95

Here is the detailed step-by-step solution for finding the squares of the given numbers using the shortcut method.

A) 35²

To find the square of 35, use the shortcut for squaring numbers near 50, which involves breaking the number into a sum of 50 and a smaller number.

Step 1: Express 35 as 50 - 15. Step 2: Square 50: 50² = 2500. Step 3: Square 15: 15² = 225. Step 4: Subtract twice the product of 50 and 15 from 2500: 2500 - 2 × 50 × 15 = 2500 - 1500 = 1000. Step 5: Add the square of 15 to the result: 1000 + 225 = 1225.

So, 35² = 1225.

B) 65²

For 65, use a similar method:

Step 1: Express 65 as 50 + 15. Step 2: Square 50: 50² = 2500. Step 3: Square 15: 15² = 225. Step 4: Add twice the product of 50 and 15 to 2500: 2500 + 2 × 50 × 15 = 2500 + 1500 = 4000. Step 5: Add the square of 15 to the result: 4000 + 225 = 4225.

So, 65² = 4225.

C) 145²

For 145, use a similar method but break it down into 100 + 45:

Step 1: Express 145 as 100 + 45. Step 2: Square 100: 100² = 10000. Step 3: Square 45: 45² = 2025. Step 4: Add twice the product of 100 and 45 to 10000: 10000 + 2 × 100 × 45 = 10000 + 9000 = 19000. Step 5: Add the square of 45 to the result: 19000 + 2025 = 21025.

So, 145² = 21025.

D) 95²

For 95, break it down into 100 - 5:

Step 1: Express 95 as 100 - 5. Step 2: Square 100: 100² = 10000. Step 3: Square 5: 5² = 25. Step 4: Subtract twice the product of 100 and 5 from 10000: 10000 - 2 × 100 × 5 = 10000 - 1000 = 9000. Step 5: Add the square of 5 to the result: 9000 + 25 = 9025.

So, 95² = 9025.

Final Answers:

A) 35² = 1225
B) 65² = 4225
C) 145² = 21025
D) 95² = 9025