Answer :
Let's evaluate the given expressions using appropriate algebraic identities.
### (i) [tex]\( (399)^2 \)[/tex]
We can use the identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex] to simplify this calculation.
For [tex]\((399)^2\)[/tex]:
- Let [tex]\( a = 400 \)[/tex] and [tex]\( b = 1 \)[/tex]
- According to the identity: [tex]\((399)^2 = (400 - 1)^2\)[/tex]
Now, let's apply the identity step-by-step:
1. [tex]\( a^2 = 400^2 = 160000 \)[/tex]
2. [tex]\( 2ab = 2 \times 400 \times 1 = 800 \)[/tex]
3. [tex]\( b^2 = 1^2 = 1 \)[/tex]
Putting these values into the identity:
[tex]\[ (399)^2 = 400^2 - 2 \times 400 \times 1 + 1^2 = 160000 - 800 + 1 = 159201 \][/tex]
So, [tex]\( (399)^2 = 159201 \)[/tex].
### (ii) [tex]\( (0.98)^2 \)[/tex]
We can use the same identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex] for this calculation as well.
For [tex]\((0.98)^2\)[/tex]:
- Let [tex]\( a = 1 \)[/tex] and [tex]\( b = 0.02 \)[/tex]
- According to the identity: [tex]\((0.98)^2 = (1 - 0.02)^2\)[/tex]
Now, let's apply the identity step-by-step:
1. [tex]\( a^2 = 1^2 = 1 \)[/tex]
2. [tex]\( 2ab = 2 \times 1 \times 0.02 = 0.04 \)[/tex]
3. [tex]\( b^2 = (0.02)^2 = 0.0004 \)[/tex]
Putting these values into the identity:
[tex]\[ (0.98)^2 = 1^2 - 2 \times 1 \times 0.02 + (0.02)^2 = 1 - 0.04 + 0.0004 = 0.9604 \][/tex]
So, [tex]\( (0.98)^2 = 0.9604 \)[/tex].
### Summary
1. [tex]\( (399)^2 = 159201 \)[/tex]
2. [tex]\( (0.98)^2 = 0.9604 \)[/tex]
These are the evaluated results using identities.
### (i) [tex]\( (399)^2 \)[/tex]
We can use the identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex] to simplify this calculation.
For [tex]\((399)^2\)[/tex]:
- Let [tex]\( a = 400 \)[/tex] and [tex]\( b = 1 \)[/tex]
- According to the identity: [tex]\((399)^2 = (400 - 1)^2\)[/tex]
Now, let's apply the identity step-by-step:
1. [tex]\( a^2 = 400^2 = 160000 \)[/tex]
2. [tex]\( 2ab = 2 \times 400 \times 1 = 800 \)[/tex]
3. [tex]\( b^2 = 1^2 = 1 \)[/tex]
Putting these values into the identity:
[tex]\[ (399)^2 = 400^2 - 2 \times 400 \times 1 + 1^2 = 160000 - 800 + 1 = 159201 \][/tex]
So, [tex]\( (399)^2 = 159201 \)[/tex].
### (ii) [tex]\( (0.98)^2 \)[/tex]
We can use the same identity [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex] for this calculation as well.
For [tex]\((0.98)^2\)[/tex]:
- Let [tex]\( a = 1 \)[/tex] and [tex]\( b = 0.02 \)[/tex]
- According to the identity: [tex]\((0.98)^2 = (1 - 0.02)^2\)[/tex]
Now, let's apply the identity step-by-step:
1. [tex]\( a^2 = 1^2 = 1 \)[/tex]
2. [tex]\( 2ab = 2 \times 1 \times 0.02 = 0.04 \)[/tex]
3. [tex]\( b^2 = (0.02)^2 = 0.0004 \)[/tex]
Putting these values into the identity:
[tex]\[ (0.98)^2 = 1^2 - 2 \times 1 \times 0.02 + (0.02)^2 = 1 - 0.04 + 0.0004 = 0.9604 \][/tex]
So, [tex]\( (0.98)^2 = 0.9604 \)[/tex].
### Summary
1. [tex]\( (399)^2 = 159201 \)[/tex]
2. [tex]\( (0.98)^2 = 0.9604 \)[/tex]
These are the evaluated results using identities.
