Using identities, evaluate. (i) `71^2` (ii) `99^2` (iii) `102^2` (iv) `998^2` (v) `5.2^2` (vi) `297 xx 303` (vii) `78 xx 82` (viii) `892` (ix) `105 xx 95`Type here in ASCII with maths in back tick:

Explanation

Let's evaluate each of the expressions step by step using algebraic identities.

### (i) Evaluate `71^2`

1. Rewrite `71` as `70 + 1`.
2. Use the identity: \((a + b)^2 = a^2 + 2ab + b^2\).
3. Substitute \(a = 70\) and \(b = 1\):
\[ 71^2 = (70 + 1)^2 = 70^2 + 2 \cdot 70 \cdot 1 + 1^2 \]
4. Calculate each term:
- \(70^2 = 4900\)
- \(2 \cdot 70 \cdot 1 = 140\)
- \(1^2 = 1\)
5. Add them together:
\[ 4900 + 140 + 1 = 5041 \]

**Answer:** \(71^2 = 5041\)

### (ii) Evaluate `99^2`

1. Rewrite `99` as `100 - 1`.
2. Use the identity: \((a - b)^2 = a^2 - 2ab + b^2\).
3. Substitute \(a = 100\) and \(b = 1\):
\[ 99^2 = (100 - 1)^2 = 100^2 - 2 \cdot 100 \cdot 1 + 1^2 \]
4. Calculate each term:
- \(100^2 = 10000\)
- \(-2 \cdot 100 \cdot 1 = -200\)
- \(1^2 = 1\)
5. Add them together:
\[ 10000 - 200 + 1 = 9801 \]

**Answer:** \(99^2 = 9801\)

### (iii) Evaluate `102^2`

1. Rewrite `102` as `100 + 2`.
2. Use the identity: \((a + b)^2 = a^2 + 2ab + b^2\).
3. Substitute \(a = 100\) and \(b = 2\):
\[ 102^2 = (100 + 2)^2 = 100^2 + 2 \cdot 100 \cdot 2 + 2^2 \]
4. Calculate each term:
- \(100^2 = 10000\)
- \(2 \cdot 100 \cdot 2 = 400\)
- \(2^2 = 4\)
5. Add them together:
\[ 10000 + 400 + 4 = 10404 \]

**Answer:** \(102^2 = 10404\)

### (iv) Evaluate `998^2`

1. Rewrite `998` as `1000 - 2`.
2. Use the identity: \((a - b)^2 = a^2 - 2ab + b^2\).
3. Substitute \(a = 1000\) and \(b = 2\):
\[ 998^2 = (1000 - 2)^2 = 1000^2 - 2 \cdot 1000 \cdot 2 + 2^2 \]
4. Calculate each term:
- \(1000^2 = 1000000\)
- \(-2 \cdot 1000 \cdot 2 = -4000\)
- \(2^2 = 4\)
5. Add them together:
\[ 1000000 - 4000 + 4 = 996004 \]

**Answer:** \(998^2 = 996004\)

### (v) Evaluate `5.2^2`

1. Rewrite `5.2` as `5 + 0.2`.
2. Use the identity: \((a + b)^2 = a^2 + 2ab + b^2\).
3. Substitute \(a = 5\) and \(b = 0.2\):
\[ 5.2^2 = (5 + 0.2)^2 = 5^2 + 2 \cdot 5 \cdot 0.2 + 0.2^2 \]
4. Calculate each term:
- \(5^2 = 25\)
- \(2 \cdot 5 \cdot 0.2 = 2\)
- \(0.2^2 = 0.04\)
5. Add them together:
\[ 25 + 2 + 0.04 = 27.04 \]

**Answer:** \(5.2^2 = 27.04\)

### (vi) Evaluate `297 x 303`

1. Rewrite `297` as `300 - 3` and `303` as `300 + 3`.
2. Use the identity: \(a^2 - b^2 = (a - b)(a + b)\).
3. Substitute \(a = 300\) and \(b = 3\):
\[ 297 \times 303 = (300 - 3)(300 + 3) = 300^2 - 3^2 \]
4. Calculate each term:
- \(300^2 = 90000\)
- \(3^2 = 9\)
5. Subtract:
\[ 90000 - 9 = 89991 \]

**Answer:** \(297 \times 303 = 89991\)

### (vii) Evaluate `78 x 82`

1. Rewrite `78` as `80 - 2` and `82` as `80 + 2`.
2. Use the identity: \(a^2 - b^2 = (a - b)(a + b)\).
3. Substitute \(a = 80\) and \(b = 2\):
\[ 78 \times 82 = (80 - 2)(80 + 2) = 80^2 - 2^2 \]
4. Calculate each term:
- \(80^2 = 6400\)
- \(2^2 = 4\)
5. Subtract:
\[ 6400 - 4 = 6396 \]

**Answer:** \(78 \times 82 = 6396\)

### (viii) Evaluate `8.9^2`

1. Rewrite `8.9` as `9 - 0.1`.
2. Use the identity: \((a - b)^2 = a^2 - 2ab + b^2\).
3. Substitute \(a = 9\) and \(b = 0.1\):
\[ 8.9^2 = (9 - 0.1)^2 = 9^2 - 2 \cdot 9 \cdot 0.1 + 0.1^2 \]
4. Calculate each term:
- \(9^2 = 81\)
- \(-2 \cdot 9 \cdot 0.1 = -1.8\)
- \(0.1^2 = 0.01\)
5. Combine:
\[ 81 - 1.8 + 0.01 = 79.21 \]

**Answer:** \(8.9^2 = 79.21\)

### (ix) Evaluate `105 x 95`

1. Rewrite `105` as `100 + 5` and `95` as `100 - 5`.
2. Use the identity: \(a^2 - b^2 = (a - b)(a + b)\).
3. Substitute \(a = 100\) and \(b = 5\):
\[ 105 \times 95 = (100 + 5)(100 - 5) = 100^2 - 5^2 \]
4. Calculate each term:
- \(100^2 = 10000\)
- \(5^2 = 25\)
5. Subtract:
\[ 10000 - 25 = 9975 \]

**Answer:** \(105 \times 95 = 9975\)

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