To find the squares of the given numbers using the formula [tex]\((a+b)(a-b) = a^2 - b^2\)[/tex], we can manipulate each number into a form that makes the calculations straightforward.
Let's calculate the squares of each number step-by-step:
### a) 49
1. Rewrite 49 as [tex]\(50 - 1\)[/tex].
2. Use the formula [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex] where [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex].
3. Calculate:
[tex]\[ (50 - 1)(50 + 1) \][/tex]
[tex]\[ 49 \cdot 51 = 50^2 - 1^2 = 2500 - 1 = 2499 \][/tex]
4. Therefore, [tex]\( 49^2 = 2499 \)[/tex].
### b) 51
1. Rewrite 51 as [tex]\(50 + 1\)[/tex].
2. Use the formula [tex]\((a+b)(a-b) = a^2 - b^2\)[/tex] where [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex].
3. Calculate:
[tex]\[ (50 + 1)(50 - 1) \][/tex]
[tex]\[ 51 \cdot 49 = 50^2 - 1^2 = 2500 - 1 = 2499 \][/tex]
4. Therefore, [tex]\( 51^2 = 2499 \)[/tex].
### c) 99
1. Rewrite 99 as [tex]\(100 - 1\)[/tex].
2. Use the formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex] where [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex].
3. Calculate:
[tex]\[ (100 - 1)(100 + 1) \][/tex]
[tex]\[ 99 \cdot 101 = 100^2 - 1^2 = 10000 - 1 = 9999 \][/tex]
4. Therefore, [tex]\( 99^2 = 9999 \)[/tex].
### d) 101
1. Rewrite 101 as [tex]\(100 + 1\)[/tex].
2. Use the formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex] where [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex].
3. Calculate:
[tex]\[ (100 + 1)(100 - 1) \][/tex]
[tex]\[ 101 \cdot 99 = 100^2 - 1^2 = 10000 - 1 = 9999 \][/tex]
4. Therefore, [tex]\( 101^2 = 9999 \)[/tex].
### e) 999
1. Rewrite 999 as [tex]\(1000 - 1\)[/tex].
2. Use the formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex] where [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex].
3. Calculate:
[tex]\[ (1000 - 1)(1000 + 1) \][/tex]
[tex]\[ 999 \cdot 1001 = 1000^2 - 1^2 = 1000000 - 1 = 999999 \][/tex]
4. Therefore, [tex]\( 999^2 = 999999 \)[/tex].
### f) 1001
1. Rewrite 1001 as [tex]\(1000 + 1\)[/tex].
2. Use the formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex] where [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex].
3. Calculate:
[tex]\[ (1000 + 1)(1000 - 1) \][/tex]
[tex]\[ 1001 \cdot 999 = 1000^2 - 1^2 = 1000000 - 1 = 999999 \][/tex]
4. Therefore, [tex]\( 1001^2 = 999999 \)[/tex].
Thus, using the formula, we find:
a) [tex]\(49^2 = 2499\)[/tex]
b) [tex]\(51^2 = 2499\)[/tex]
c) [tex]\(99^2 = 9999\)[/tex]
d) [tex]\(101^2 = 9999\)[/tex]
e) [tex]\(999^2 = 999999\)[/tex]
f) [tex]\(1001^2 = 999999\)[/tex]
