Answer :
Certainly! Let's find the squares of the given numbers using the formulas for [tex]\((a+b)^2\)[/tex] and [tex]\((a-b)^2\)[/tex].
### a) 49
To find the square of 49, we can represent it as [tex]\((50 - 1)^2\)[/tex]. This fits the formula for [tex]\((a - b)^2\)[/tex], where [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
When [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (50 - 1)^2 = 50^2 - 2 \cdot 50 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 2500 - 100 + 1 \][/tex]
[tex]\[ = 2401 \][/tex]
Thus, [tex]\(49^2 = 2401\)[/tex].
### b) 51
To find the square of 51, we can represent it as [tex]\((50 + 1)^2\)[/tex]. This fits the formula for [tex]\((a + b)^2\)[/tex], where [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
When [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (50 + 1)^2 = 50^2 + 2 \cdot 50 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 2500 + 100 + 1 \][/tex]
[tex]\[ = 2601 \][/tex]
Thus, [tex]\(51^2 = 2601\)[/tex].
### c) 99
To find the square of 99, we can represent it as [tex]\((100 - 1)^2\)[/tex]. This fits the formula for [tex]\((a - b)^2\)[/tex], where [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
When [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (100 - 1)^2 = 100^2 - 2 \cdot 100 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 10000 - 200 + 1 \][/tex]
[tex]\[ = 9801 \][/tex]
Thus, [tex]\(99^2 = 9801\)[/tex].
### d) 101
To find the square of 101, we can represent it as [tex]\((100 + 1)^2\)[/tex]. This fits the formula for [tex]\((a + b)^2\)[/tex], where [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
When [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (100 + 1)^2 = 100^2 + 2 \cdot 100 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 10000 + 200 + 1 \][/tex]
[tex]\[ = 10201 \][/tex]
Thus, [tex]\(101^2 = 10201\)[/tex].
### e) 999
To find the square of 999, we can represent it as [tex]\((1000 - 1)^2\)[/tex]. This fits the formula for [tex]\((a - b)^2\)[/tex], where [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
When [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (1000 - 1)^2 = 1000^2 - 2 \cdot 1000 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 1000000 - 2000 + 1 \][/tex]
[tex]\[ = 998001 \][/tex]
Thus, [tex]\(999^2 = 998001\)[/tex].
### f) 1001
To find the square of 1001, we can represent it as [tex]\((1000 + 1)^2\)[/tex]. This fits the formula for [tex]\((a + b)^2\)[/tex], where [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
When [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (1000 + 1)^2 = 1000^2 + 2 \cdot 1000 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 1000000 + 2000 + 1 \][/tex]
[tex]\[ = 1002001 \][/tex]
Thus, [tex]\(1001^2 = 1002001\)[/tex].
### a) 49
To find the square of 49, we can represent it as [tex]\((50 - 1)^2\)[/tex]. This fits the formula for [tex]\((a - b)^2\)[/tex], where [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
When [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (50 - 1)^2 = 50^2 - 2 \cdot 50 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 2500 - 100 + 1 \][/tex]
[tex]\[ = 2401 \][/tex]
Thus, [tex]\(49^2 = 2401\)[/tex].
### b) 51
To find the square of 51, we can represent it as [tex]\((50 + 1)^2\)[/tex]. This fits the formula for [tex]\((a + b)^2\)[/tex], where [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
When [tex]\(a = 50\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (50 + 1)^2 = 50^2 + 2 \cdot 50 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 2500 + 100 + 1 \][/tex]
[tex]\[ = 2601 \][/tex]
Thus, [tex]\(51^2 = 2601\)[/tex].
### c) 99
To find the square of 99, we can represent it as [tex]\((100 - 1)^2\)[/tex]. This fits the formula for [tex]\((a - b)^2\)[/tex], where [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
When [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (100 - 1)^2 = 100^2 - 2 \cdot 100 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 10000 - 200 + 1 \][/tex]
[tex]\[ = 9801 \][/tex]
Thus, [tex]\(99^2 = 9801\)[/tex].
### d) 101
To find the square of 101, we can represent it as [tex]\((100 + 1)^2\)[/tex]. This fits the formula for [tex]\((a + b)^2\)[/tex], where [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
When [tex]\(a = 100\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (100 + 1)^2 = 100^2 + 2 \cdot 100 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 10000 + 200 + 1 \][/tex]
[tex]\[ = 10201 \][/tex]
Thus, [tex]\(101^2 = 10201\)[/tex].
### e) 999
To find the square of 999, we can represent it as [tex]\((1000 - 1)^2\)[/tex]. This fits the formula for [tex]\((a - b)^2\)[/tex], where [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
When [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (1000 - 1)^2 = 1000^2 - 2 \cdot 1000 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 1000000 - 2000 + 1 \][/tex]
[tex]\[ = 998001 \][/tex]
Thus, [tex]\(999^2 = 998001\)[/tex].
### f) 1001
To find the square of 1001, we can represent it as [tex]\((1000 + 1)^2\)[/tex]. This fits the formula for [tex]\((a + b)^2\)[/tex], where [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex].
Applying the formula:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
When [tex]\(a = 1000\)[/tex] and [tex]\(b = 1\)[/tex]:
[tex]\[ (1000 + 1)^2 = 1000^2 + 2 \cdot 1000 \cdot 1 + 1^2 \][/tex]
[tex]\[ = 1000000 + 2000 + 1 \][/tex]
[tex]\[ = 1002001 \][/tex]
Thus, [tex]\(1001^2 = 1002001\)[/tex].
