Find the value of:
(i) `(82)^(2) - (18)^(2)` (ii) `(128)^(2) - (72)^(2)` (iii) `197 xx 203`
(iv) `(198 xx 198 - 102 xx 102)/(96)` (v) `(14.7 xx 15.3)` (vi) `(8.63)^(2) - (1.37)^(2)`

Answer

Let's solve each part of the question step by step.

### (i) Find the value of \( (82)^2 - (18)^2 \)

**Step 1:** Recognize that this expression is in the form of \( a^2 - b^2 \), which can be factored using the difference of squares formula:
\[ a^2 - b^2 = (a - b)(a + b) \]

**Step 2:** Here, \( a = 82 \) and \( b = 18 \).
So, we calculate:
\[ a - b = 82 - 18 = 64 \]
\[ a + b = 82 + 18 = 100 \]

**Step 3:** Substitute these values into the factored form:
\[ (82)^2 - (18)^2 = (64)(100) \]

**Step 4:** Calculate the product:
\[ 64 \times 100 = 6400 \]

**Final Answer for (i):** \( 6400 \)

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### (ii) Find the value of \( (128)^2 - (72)^2 \)

**Step 1:** Again, recognize the difference of squares:
\[ a^2 - b^2 = (a - b)(a + b) \]

**Step 2:** Here, \( a = 128 \) and \( b = 72 \).
So, we calculate:
\[ a - b = 128 - 72 = 56 \]
\[ a + b = 128 + 72 = 200 \]

**Step 3:** Substitute these values into the factored form:
\[ (128)^2 - (72)^2 = (56)(200) \]

**Step 4:** Calculate the product:
\[ 56 \times 200 = 11200 \]

**Final Answer for (ii):** \( 11200 \)

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### (iii) Find the value of \( 197 \times 203 \)

**Step 1:** Recognize that this can be expressed in the form \( (a - b)(a + b) \) where \( a = 200 \) and \( b = 3 \):
\[ 197 = 200 - 3 \]
\[ 203 = 200 + 3 \]

**Step 2:** Use the difference of squares formula:
\[ 197 \times 203 = (200 - 3)(200 + 3) = 200^2 - 3^2 \]

**Step 3:** Calculate \( 200^2 \) and \( 3^2 \):
\[ 200^2 = 40000 \]
\[ 3^2 = 9 \]

**Step 4:** Substitute these values:
\[ 197 \times 203 = 40000 - 9 = 39991 \]

**Final Answer for (iii):** \( 39991 \)

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### (iv) Find the value of \( \frac{(198 \times 198) - (102 \times 102)}{96} \)

**Step 1:** Recognize the numerator as a difference of squares:
\[ (198)^2 - (102)^2 = (198 - 102)(198 + 102) \]

**Step 2:** Calculate \( 198 - 102 \) and \( 198 + 102 \):
\[ 198 - 102 = 96 \]
\[ 198 + 102 = 300 \]

**Step 3:** Substitute these values into the formula:
\[ (198)^2 - (102)^2 = (96)(300) \]

**Step 4:** Now substitute into the original expression:
\[ \frac{(96)(300)}{96} = 300 \]

**Final Answer for (iv):** \( 300 \)

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### (v) Find the value of \( 14.7 \times 15.3 \)

**Step 1:** Recognize that this can be expressed as:
\[ 14.7 \times 15.3 = (15 - 0.3)(15 + 0.3) \]

**Step 2:** Use the difference of squares formula:
\[ 14.7 \times 15.3 = 15^2 - (0.3)^2 \]

**Step 3:** Calculate \( 15^2 \) and \( (0.3)^2 \):
\[ 15^2 = 225 \]
\[ (0.3)^2 = 0.09 \]

**Step 4:** Substitute these values:
\[ 14.7 \times 15.3 = 225 - 0.09 = 224.91 \]

**Final Answer for (v):** \( 224.91 \)

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### (vi) Find the value of \( (8.63)^2 - (1.37)^2 \)

**Step 1:** Recognize this as a difference of squares:
\[ (8.63)^2 - (1.37)^2 = (8.63 - 1.37)(8.63 + 1.37) \]

**Step 2:** Calculate \( 8.63 - 1.37 \) and \( 8.63 + 1.37 \):
\[ 8.63 - 1.37 = 7.26 \]
\[ 8.63 + 1.37 = 10 \]

**Step 3:** Substitute these values into the formula:
\[ (8.63)^2 - (1.37)^2 = (7.26)(10) \]

**Step 4:** Calculate the product:
\[ 7.26 \times 10 = 72.6 \]

**Final Answer for (vi):** \( 72.6 \)

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