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Answer :
Sure! Let's break down each part of the problem using algebra:
### Part (a): [tex]\(268^2 - 232^2\)[/tex]
To solve this, we can use the difference of squares formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Let's set [tex]\( a = 268 \)[/tex] and [tex]\( b = 232 \)[/tex].
1. Calculate [tex]\( a - b \)[/tex]:
[tex]\[ 268 - 232 = 36 \][/tex]
2. Calculate [tex]\( a + b \)[/tex]:
[tex]\[ 268 + 232 = 500 \][/tex]
3. Apply the difference of squares formula:
[tex]\[ 268^2 - 232^2 = (268 - 232)(268 + 232) = 36 \times 500 = 18000 \][/tex]
So, the result for part (a) is 18000.
### Part (b): [tex]\(469 \times 548 + 469^2 - 469 \times 17\)[/tex]
Here, we can factor by grouping:
1. Notice that [tex]\(469\)[/tex] is common in all terms, so let's factor [tex]\(469\)[/tex] out:
[tex]\[ 469 \times (548 + 469 - 17) \][/tex]
2. Simplify inside the parentheses:
[tex]\[ 548 + 469 - 17 = 1000 \][/tex]
3. Multiply:
[tex]\[ 469 \times 1000 = 469000 \][/tex]
So, the result for part (b) is 469000.
### Part (c): [tex]\(\frac{65.1 \times 29.2 + 65.1 \times 35.9 - 91.7 \times 26.4 + 65.3 \times 26.4}{18.3^2 - 18.3 \times 5.4}\)[/tex]
For this part, we simplify both the numerator and the denominator:
1. Factor and simplify the numerator:
- Notice that [tex]\(65.1\)[/tex] is common in the first two terms:
[tex]\[ 65.1 \times (29.2 + 35.9) - 91.7 \times 26.4 + 65.3 \times 26.4 \][/tex]
- Simplify inside the parenthesis:
[tex]\[ 29.2 + 35.9 = 65.1 \][/tex]
- This simplifies further to:
[tex]\[ 65.1 \times 65.1 - 91.7 \times 26.4 + 65.3 \times 26.4 \][/tex]
- Combine like terms for the [tex]\(26.4\)[/tex] terms:
[tex]\[ 65.1 \times 65.1 + (65.3 - 91.7) \times 26.4 \][/tex]
2. Simplify the denominator:
- Use distribution:
[tex]\[ 18.3^2 - 18.3 \times 5.4 \][/tex]
[tex]\[ 18.3 \times (18.3 - 5.4) = 18.3 \times 12.9 \][/tex]
3. Calculate the result:
[tex]\[ \text{Result} = \frac{\text{Simplified Numerator}}{\text{Simplified Denominator}} = 15 \][/tex]
Therefore, the result for part (c) is approximately 15.
### Part (a): [tex]\(268^2 - 232^2\)[/tex]
To solve this, we can use the difference of squares formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Let's set [tex]\( a = 268 \)[/tex] and [tex]\( b = 232 \)[/tex].
1. Calculate [tex]\( a - b \)[/tex]:
[tex]\[ 268 - 232 = 36 \][/tex]
2. Calculate [tex]\( a + b \)[/tex]:
[tex]\[ 268 + 232 = 500 \][/tex]
3. Apply the difference of squares formula:
[tex]\[ 268^2 - 232^2 = (268 - 232)(268 + 232) = 36 \times 500 = 18000 \][/tex]
So, the result for part (a) is 18000.
### Part (b): [tex]\(469 \times 548 + 469^2 - 469 \times 17\)[/tex]
Here, we can factor by grouping:
1. Notice that [tex]\(469\)[/tex] is common in all terms, so let's factor [tex]\(469\)[/tex] out:
[tex]\[ 469 \times (548 + 469 - 17) \][/tex]
2. Simplify inside the parentheses:
[tex]\[ 548 + 469 - 17 = 1000 \][/tex]
3. Multiply:
[tex]\[ 469 \times 1000 = 469000 \][/tex]
So, the result for part (b) is 469000.
### Part (c): [tex]\(\frac{65.1 \times 29.2 + 65.1 \times 35.9 - 91.7 \times 26.4 + 65.3 \times 26.4}{18.3^2 - 18.3 \times 5.4}\)[/tex]
For this part, we simplify both the numerator and the denominator:
1. Factor and simplify the numerator:
- Notice that [tex]\(65.1\)[/tex] is common in the first two terms:
[tex]\[ 65.1 \times (29.2 + 35.9) - 91.7 \times 26.4 + 65.3 \times 26.4 \][/tex]
- Simplify inside the parenthesis:
[tex]\[ 29.2 + 35.9 = 65.1 \][/tex]
- This simplifies further to:
[tex]\[ 65.1 \times 65.1 - 91.7 \times 26.4 + 65.3 \times 26.4 \][/tex]
- Combine like terms for the [tex]\(26.4\)[/tex] terms:
[tex]\[ 65.1 \times 65.1 + (65.3 - 91.7) \times 26.4 \][/tex]
2. Simplify the denominator:
- Use distribution:
[tex]\[ 18.3^2 - 18.3 \times 5.4 \][/tex]
[tex]\[ 18.3 \times (18.3 - 5.4) = 18.3 \times 12.9 \][/tex]
3. Calculate the result:
[tex]\[ \text{Result} = \frac{\text{Simplified Numerator}}{\text{Simplified Denominator}} = 15 \][/tex]
Therefore, the result for part (c) is approximately 15.
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