Answer :
Absolutely, let's simplify each of these expressions using the formula [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex]:
### (i) [tex]\((82)^2 - (18)^2\)[/tex]
Using the formula:
[tex]\[ (82)^2 - (18)^2 = (82 - 18)(82 + 18) \][/tex]
Calculating the values:
[tex]\[ 82 - 18 = 64 \quad \text{and} \quad 82 + 18 = 100 \][/tex]
Thus:
[tex]\[ (82)^2 - (18)^2 = 64 \times 100 = 6400 \][/tex]
### (ii) [tex]\((467)^2 - (33)^2\)[/tex]
Using the formula:
[tex]\[ (467)^2 - (33)^2 = (467 - 33)(467 + 33) \][/tex]
Calculating the values:
[tex]\[ 467 - 33 = 434 \quad \text{and} \quad 467 + 33 = 500 \][/tex]
Thus:
[tex]\[ (467)^2 - (33)^2 = 434 \times 500 = 217000 \][/tex]
### (iii) [tex]\((79)^2 - (69)^2\)[/tex]
Using the formula:
[tex]\[ (79)^2 - (69)^2 = (79 - 69)(79 + 69) \][/tex]
Calculating the values:
[tex]\[ 79 - 69 = 10 \quad \text{and} \quad 79 + 69 = 148 \][/tex]
Thus:
[tex]\[ (79)^2 - (69)^2 = 10 \times 148 = 1480 \][/tex]
### (iv) [tex]\(197 \times 203\)[/tex]
Here we rewrite in the form [tex]\( (200-3)(200+3) \)[/tex]:
[tex]\[ 197 \times 203 = (200 - 3)(200 + 3) \][/tex]
Using the formula:
[tex]\[ (200 - 3)(200 + 3) = 200^2 - 3^2 \][/tex]
Calculating the values:
[tex]\[ 200^2 = 40000 \quad \text{and} \quad 3^2 = 9 \][/tex]
Thus:
[tex]\[ 197 \times 203 = 40000 - 9 = 39991 \][/tex]
### (v) [tex]\(113 \times 87\)[/tex]
Here we rewrite in the form [tex]\( (100 + 13)(100 - 13) \)[/tex]:
[tex]\[ 113 \times 87 = (100 + 13)(100 - 13) \][/tex]
Using the formula:
[tex]\[ (100 + 13)(100 - 13) = 100^2 - 13^2 \][/tex]
Calculating the values:
[tex]\[ 100^2 = 10000 \quad \text{and} \quad 13^2 = 169 \][/tex]
Thus:
[tex]\[ 113 \times 87 = 10000 - 169 = 9831 \][/tex]
### (vi) [tex]\(95 \times 105\)[/tex]
Here we rewrite in the form [tex]\( (100 - 5)(100 + 5) \)[/tex]:
[tex]\[ 95 \times 105 = (100 - 5)(100 + 5) \][/tex]
Using the formula:
[tex]\[ (100 - 5)(100 + 5) = 100^2 - 5^2 \][/tex]
Calculating the values:
[tex]\[ 100^2 = 10000 \quad \text{and} \quad 5^2 = 25 \][/tex]
Thus:
[tex]\[ 95 \times 105 = 10000 - 25 = 9975 \][/tex]
### (vii) [tex]\(1.8 \times 2.2\)[/tex]
Here we rewrite in the form [tex]\( (2 - 0.2)(2 + 0.2) \)[/tex]:
[tex]\[ 1.8 \times 2.2 = (2 - 0.2)(2 + 0.2) \][/tex]
Using the formula:
[tex]\[ (2 - 0.2)(2 + 0.2) = 2^2 - 0.2^2 \][/tex]
Calculating the values:
[tex]\[ 2^2 = 4 \quad \text{and} \quad 0.2^2 = 0.04 \][/tex]
Thus:
[tex]\[ 1.8 \times 2.2 = 4 - 0.04 = 3.96 \][/tex]
### (viii) [tex]\(9.8 \times 10.2\)[/tex]
Here we rewrite in the form [tex]\( (10 - 0.2)(10 + 0.2) \)[/tex]:
[tex]\[ 9.8 \times 10.2 = (10 - 0.2)(10 + 0.2) \][/tex]
Using the formula:
[tex]\[ (10 - 0.2)(10 + 0.2) = 10^2 - 0.2^2 \][/tex]
Calculating the values:
[tex]\[ 10^2 = 100 \quad \text{and} \quad 0.2^2 = 0.04 \][/tex]
Thus:
[tex]\[ 9.8 \times 10.2 = 100 - 0.04 = 99.96 \][/tex]
So, the simplified results are:
(i) 6400
(ii) 217000
(iii) 1480
(iv) 39991
(v) 9831
(vi) 9975
(vii) 3.96
(viii) 99.96
### (i) [tex]\((82)^2 - (18)^2\)[/tex]
Using the formula:
[tex]\[ (82)^2 - (18)^2 = (82 - 18)(82 + 18) \][/tex]
Calculating the values:
[tex]\[ 82 - 18 = 64 \quad \text{and} \quad 82 + 18 = 100 \][/tex]
Thus:
[tex]\[ (82)^2 - (18)^2 = 64 \times 100 = 6400 \][/tex]
### (ii) [tex]\((467)^2 - (33)^2\)[/tex]
Using the formula:
[tex]\[ (467)^2 - (33)^2 = (467 - 33)(467 + 33) \][/tex]
Calculating the values:
[tex]\[ 467 - 33 = 434 \quad \text{and} \quad 467 + 33 = 500 \][/tex]
Thus:
[tex]\[ (467)^2 - (33)^2 = 434 \times 500 = 217000 \][/tex]
### (iii) [tex]\((79)^2 - (69)^2\)[/tex]
Using the formula:
[tex]\[ (79)^2 - (69)^2 = (79 - 69)(79 + 69) \][/tex]
Calculating the values:
[tex]\[ 79 - 69 = 10 \quad \text{and} \quad 79 + 69 = 148 \][/tex]
Thus:
[tex]\[ (79)^2 - (69)^2 = 10 \times 148 = 1480 \][/tex]
### (iv) [tex]\(197 \times 203\)[/tex]
Here we rewrite in the form [tex]\( (200-3)(200+3) \)[/tex]:
[tex]\[ 197 \times 203 = (200 - 3)(200 + 3) \][/tex]
Using the formula:
[tex]\[ (200 - 3)(200 + 3) = 200^2 - 3^2 \][/tex]
Calculating the values:
[tex]\[ 200^2 = 40000 \quad \text{and} \quad 3^2 = 9 \][/tex]
Thus:
[tex]\[ 197 \times 203 = 40000 - 9 = 39991 \][/tex]
### (v) [tex]\(113 \times 87\)[/tex]
Here we rewrite in the form [tex]\( (100 + 13)(100 - 13) \)[/tex]:
[tex]\[ 113 \times 87 = (100 + 13)(100 - 13) \][/tex]
Using the formula:
[tex]\[ (100 + 13)(100 - 13) = 100^2 - 13^2 \][/tex]
Calculating the values:
[tex]\[ 100^2 = 10000 \quad \text{and} \quad 13^2 = 169 \][/tex]
Thus:
[tex]\[ 113 \times 87 = 10000 - 169 = 9831 \][/tex]
### (vi) [tex]\(95 \times 105\)[/tex]
Here we rewrite in the form [tex]\( (100 - 5)(100 + 5) \)[/tex]:
[tex]\[ 95 \times 105 = (100 - 5)(100 + 5) \][/tex]
Using the formula:
[tex]\[ (100 - 5)(100 + 5) = 100^2 - 5^2 \][/tex]
Calculating the values:
[tex]\[ 100^2 = 10000 \quad \text{and} \quad 5^2 = 25 \][/tex]
Thus:
[tex]\[ 95 \times 105 = 10000 - 25 = 9975 \][/tex]
### (vii) [tex]\(1.8 \times 2.2\)[/tex]
Here we rewrite in the form [tex]\( (2 - 0.2)(2 + 0.2) \)[/tex]:
[tex]\[ 1.8 \times 2.2 = (2 - 0.2)(2 + 0.2) \][/tex]
Using the formula:
[tex]\[ (2 - 0.2)(2 + 0.2) = 2^2 - 0.2^2 \][/tex]
Calculating the values:
[tex]\[ 2^2 = 4 \quad \text{and} \quad 0.2^2 = 0.04 \][/tex]
Thus:
[tex]\[ 1.8 \times 2.2 = 4 - 0.04 = 3.96 \][/tex]
### (viii) [tex]\(9.8 \times 10.2\)[/tex]
Here we rewrite in the form [tex]\( (10 - 0.2)(10 + 0.2) \)[/tex]:
[tex]\[ 9.8 \times 10.2 = (10 - 0.2)(10 + 0.2) \][/tex]
Using the formula:
[tex]\[ (10 - 0.2)(10 + 0.2) = 10^2 - 0.2^2 \][/tex]
Calculating the values:
[tex]\[ 10^2 = 100 \quad \text{and} \quad 0.2^2 = 0.04 \][/tex]
Thus:
[tex]\[ 9.8 \times 10.2 = 100 - 0.04 = 99.96 \][/tex]
So, the simplified results are:
(i) 6400
(ii) 217000
(iii) 1480
(iv) 39991
(v) 9831
(vi) 9975
(vii) 3.96
(viii) 99.96
