Answer :
Let's evaluate the given expressions using mathematical identities.
### (i) [tex]\((399)^2\)[/tex]
To find [tex]\((399)^2\)[/tex], we can use the identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. We choose a nearby number to 399 that is easier to square. In this case, we take [tex]\(a = 400\)[/tex], and since [tex]\(399 = 400 - 1\)[/tex], we have [tex]\(b = 1\)[/tex].
Using the identity:
[tex]\[ (399)^2 = (400 - 1)^2 \][/tex]
Substitute into the identity:
[tex]\[ (400 - 1)^2 = 400^2 - 2 \cdot 400 \cdot 1 + 1^2 \][/tex]
Calculate each term separately:
1. [tex]\(400^2 = 160000\)[/tex]
2. [tex]\(2 \cdot 400 \cdot 1 = 800\)[/tex]
3. [tex]\(1^2 = 1\)[/tex]
Putting it all together:
[tex]\[ 400^2 - 2 \cdot 400 \cdot 1 + 1^2 = 160000 - 800 + 1 = 159201 \][/tex]
Therefore, [tex]\((399)^2 = 159201\)[/tex].
### (ii) [tex]\((0.98)^2\)[/tex]
To find [tex]\((0.98)^2\)[/tex], we use the same identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. Here, we choose [tex]\(a = 1.0\)[/tex] (since it is close to 0.98) and [tex]\(b = 0.02\)[/tex] (because [tex]\(0.98 = 1.0 - 0.02\)[/tex]).
Using the identity, we have:
[tex]\[ (0.98)^2 = (1.0 - 0.02)^2 \][/tex]
Substitute into the identity:
[tex]\[ (1.0 - 0.02)^2 = 1.0^2 - 2 \cdot 1.0 \cdot 0.02 + (0.02)^2 \][/tex]
Calculate each term separately:
1. [tex]\(1.0^2 = 1.0\)[/tex]
2. [tex]\(2 \cdot 1.0 \cdot 0.02 = 0.04\)[/tex]
3. [tex]\((0.02)^2 = 0.0004\)[/tex]
Putting it all together:
[tex]\[ 1.0^2 - 2 \cdot 1.0 \cdot 0.02 + (0.02)^2 = 1.0 - 0.04 + 0.0004 = 0.9604 \][/tex]
Therefore, [tex]\((0.98)^2 = 0.9604\)[/tex].
So, the evaluated results are:
(i) [tex]\((399)^2 = 159201\)[/tex]
(ii) [tex]\((0.98)^2 = 0.9604\)[/tex]
### (i) [tex]\((399)^2\)[/tex]
To find [tex]\((399)^2\)[/tex], we can use the identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. We choose a nearby number to 399 that is easier to square. In this case, we take [tex]\(a = 400\)[/tex], and since [tex]\(399 = 400 - 1\)[/tex], we have [tex]\(b = 1\)[/tex].
Using the identity:
[tex]\[ (399)^2 = (400 - 1)^2 \][/tex]
Substitute into the identity:
[tex]\[ (400 - 1)^2 = 400^2 - 2 \cdot 400 \cdot 1 + 1^2 \][/tex]
Calculate each term separately:
1. [tex]\(400^2 = 160000\)[/tex]
2. [tex]\(2 \cdot 400 \cdot 1 = 800\)[/tex]
3. [tex]\(1^2 = 1\)[/tex]
Putting it all together:
[tex]\[ 400^2 - 2 \cdot 400 \cdot 1 + 1^2 = 160000 - 800 + 1 = 159201 \][/tex]
Therefore, [tex]\((399)^2 = 159201\)[/tex].
### (ii) [tex]\((0.98)^2\)[/tex]
To find [tex]\((0.98)^2\)[/tex], we use the same identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. Here, we choose [tex]\(a = 1.0\)[/tex] (since it is close to 0.98) and [tex]\(b = 0.02\)[/tex] (because [tex]\(0.98 = 1.0 - 0.02\)[/tex]).
Using the identity, we have:
[tex]\[ (0.98)^2 = (1.0 - 0.02)^2 \][/tex]
Substitute into the identity:
[tex]\[ (1.0 - 0.02)^2 = 1.0^2 - 2 \cdot 1.0 \cdot 0.02 + (0.02)^2 \][/tex]
Calculate each term separately:
1. [tex]\(1.0^2 = 1.0\)[/tex]
2. [tex]\(2 \cdot 1.0 \cdot 0.02 = 0.04\)[/tex]
3. [tex]\((0.02)^2 = 0.0004\)[/tex]
Putting it all together:
[tex]\[ 1.0^2 - 2 \cdot 1.0 \cdot 0.02 + (0.02)^2 = 1.0 - 0.04 + 0.0004 = 0.9604 \][/tex]
Therefore, [tex]\((0.98)^2 = 0.9604\)[/tex].
So, the evaluated results are:
(i) [tex]\((399)^2 = 159201\)[/tex]
(ii) [tex]\((0.98)^2 = 0.9604\)[/tex]
