Sure, let's solve these problems step-by-step using algebraic identities.
### Part i) [tex]\(\frac{7.83 \times 7.83 - 1.17 \times 1.17}{6.66}\)[/tex]
To simplify this expression, we can use the identity:
[tex]\[a^2 - b^2 = (a + b)(a - b)\][/tex]
Here, [tex]\(a = 7.83\)[/tex] and [tex]\(b = 1.17\)[/tex].
First, calculate the numerator:
[tex]\[
a^2 - b^2 = 7.83^2 - 1.17^2
\][/tex]
Using the identity:
[tex]\[
a^2 - b^2 = (7.83 + 1.17)(7.83 - 1.17)
\][/tex]
Calculate each part separately:
[tex]\[
7.83 + 1.17 = 9.00
\][/tex]
[tex]\[
7.83 - 1.17 = 6.66
\][/tex]
Now substitute back into the identity:
[tex]\[
7.83^2 - 1.17^2 = 9.00 \times 6.66
\][/tex]
Now we plug this back into the original expression:
[tex]\[
\frac{7.83^2 - 1.17^2}{6.66} = \frac{9.00 \times 6.66}{6.66}
\][/tex]
The [tex]\(6.66\)[/tex] terms cancel out:
[tex]\[
\frac{9.00 \times 6.66}{6.66} = 9.00
\][/tex]
So, the result is:
[tex]\[
9.0
\][/tex]
### Part ii) [tex]\(29^3 - 11^3\)[/tex]
We can use the difference of cubes identity, which states:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
Here, [tex]\(a = 29\)[/tex] and [tex]\(b = 11\)[/tex].
First, compute [tex]\(a - b\)[/tex]:
[tex]\[
29 - 11 = 18
\][/tex]
Next, compute [tex]\(a^2\)[/tex], [tex]\(ab\)[/tex], and [tex]\(b^2\)[/tex]:
[tex]\[
a^2 = 29^2 = 841
\][/tex]
[tex]\[
ab = 29 \times 11 = 319
\][/tex]
[tex]\[
b^2 = 11^2 = 121
\][/tex]
Now, add these values together:
[tex]\[
a^2 + ab + b^2 = 841 + 319 + 121 = 1281
\][/tex]
Using the difference of cubes identity:
[tex]\[
29^3 - 11^3 = (29 - 11)(29^2 + 29 \times 11 + 11^2)
\][/tex]
[tex]\[
29^3 - 11^3 = 18 \times 1281
\][/tex]
Finally, calculate the result:
[tex]\[
18 \times 1281 = 23058
\][/tex]
So, the result for the second part is:
[tex]\[
23058
\][/tex]
In summary:
i) [tex]\(\frac{7.83 \times 7.83 - 1.17 \times 1.17}{6.66} = 9.0\)[/tex]
ii) [tex]\(29^3 - 11^3 = 23058\)[/tex]
