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Answer :
Sure! Let's solve these problems step-by-step using algebraic techniques:
a) [tex]\(268^2 - 232^2\)[/tex]:
For this expression, we can use the difference of squares formula, which states:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(a = 268\)[/tex] and [tex]\(b = 232\)[/tex]. So:
1. Calculate [tex]\(a - b\)[/tex]:
[tex]\[
268 - 232 = 36
\][/tex]
2. Calculate [tex]\(a + b\)[/tex]:
[tex]\[
268 + 232 = 500
\][/tex]
3. Use the difference of squares formula:
[tex]\[
268^2 - 232^2 = (268 - 232)(268 + 232) = 36 \times 500 = 18000
\][/tex]
So, the answer is [tex]\(18000\)[/tex].
b) [tex]\(469 \times 548 + 469^2 - 469 \times 17\)[/tex]:
We can factor out the common term [tex]\(469\)[/tex]:
1. Recognize the common factor:
[tex]\[
469 \times (548 + 469 - 17)
\][/tex]
2. Simplify inside the parentheses:
[tex]\[
548 + 469 - 17 = 1000
\][/tex]
3. Calculate:
[tex]\[
469 \times 1000 = 469000
\][/tex]
So, the answer is [tex]\(469000\)[/tex].
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]:
Let's simplify the numerator and the denominator separately:
1. Numerator:
Group and factor similar terms:
[tex]\[
65.1 \times (29.2 + 35.9) + 26.4 \times (65.3 - 91.7)
\][/tex]
- Simplify [tex]\(29.2 + 35.9\)[/tex]:
[tex]\[
29.2 + 35.9 = 65.1
\][/tex]
- Simplify [tex]\(65.3 - 91.7\)[/tex]:
[tex]\[
65.3 - 91.7 = -26.4
\][/tex]
- Substitute back:
[tex]\[
65.1 \times 65.1 + 26.4 \times (-26.4)
\][/tex]
Calculate:
[tex]\[
65.1^2 - 26.4^2
\][/tex]
2. Denominator:
Use basic algebraic simplification:
[tex]\[
18.3^2 - 18.3 \times 5.4 = 18.3 \times (18.3 - 5.4)
\][/tex]
- Simplify [tex]\(18.3 - 5.4\)[/tex]:
[tex]\[
18.3 - 5.4 = 12.9
\][/tex]
Calculate:
[tex]\[
18.3 \times 12.9
\][/tex]
3. Combine the results:
The simplified expression:
[tex]\[
\frac{65.1^2 - 26.4^2}{18.3 \times 12.9}
\][/tex]
By solving the simplified components, the final result of the expression is approximately [tex]\(15\)[/tex].
So, the answer for part c is [tex]\(15\)[/tex].
Putting it all together, the solutions are:
- a) 18000
- b) 469000
- c) 15
a) [tex]\(268^2 - 232^2\)[/tex]:
For this expression, we can use the difference of squares formula, which states:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(a = 268\)[/tex] and [tex]\(b = 232\)[/tex]. So:
1. Calculate [tex]\(a - b\)[/tex]:
[tex]\[
268 - 232 = 36
\][/tex]
2. Calculate [tex]\(a + b\)[/tex]:
[tex]\[
268 + 232 = 500
\][/tex]
3. Use the difference of squares formula:
[tex]\[
268^2 - 232^2 = (268 - 232)(268 + 232) = 36 \times 500 = 18000
\][/tex]
So, the answer is [tex]\(18000\)[/tex].
b) [tex]\(469 \times 548 + 469^2 - 469 \times 17\)[/tex]:
We can factor out the common term [tex]\(469\)[/tex]:
1. Recognize the common factor:
[tex]\[
469 \times (548 + 469 - 17)
\][/tex]
2. Simplify inside the parentheses:
[tex]\[
548 + 469 - 17 = 1000
\][/tex]
3. Calculate:
[tex]\[
469 \times 1000 = 469000
\][/tex]
So, the answer is [tex]\(469000\)[/tex].
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]:
Let's simplify the numerator and the denominator separately:
1. Numerator:
Group and factor similar terms:
[tex]\[
65.1 \times (29.2 + 35.9) + 26.4 \times (65.3 - 91.7)
\][/tex]
- Simplify [tex]\(29.2 + 35.9\)[/tex]:
[tex]\[
29.2 + 35.9 = 65.1
\][/tex]
- Simplify [tex]\(65.3 - 91.7\)[/tex]:
[tex]\[
65.3 - 91.7 = -26.4
\][/tex]
- Substitute back:
[tex]\[
65.1 \times 65.1 + 26.4 \times (-26.4)
\][/tex]
Calculate:
[tex]\[
65.1^2 - 26.4^2
\][/tex]
2. Denominator:
Use basic algebraic simplification:
[tex]\[
18.3^2 - 18.3 \times 5.4 = 18.3 \times (18.3 - 5.4)
\][/tex]
- Simplify [tex]\(18.3 - 5.4\)[/tex]:
[tex]\[
18.3 - 5.4 = 12.9
\][/tex]
Calculate:
[tex]\[
18.3 \times 12.9
\][/tex]
3. Combine the results:
The simplified expression:
[tex]\[
\frac{65.1^2 - 26.4^2}{18.3 \times 12.9}
\][/tex]
By solving the simplified components, the final result of the expression is approximately [tex]\(15\)[/tex].
So, the answer for part c is [tex]\(15\)[/tex].
Putting it all together, the solutions are:
- a) 18000
- b) 469000
- c) 15
Thank you for reading the article Show how algebra can be used to work out each of these without a calculator a tex 268 2 232 2 tex b tex 469. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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