To evaluate the expression [tex]\(\frac{7.83 \times 7.83 - 1.17 \times 1.17}{6.66}\)[/tex] using identities, follow these steps:
1. Recognize the Identity to Use:
The expression in the numerator, [tex]\(7.83 \times 7.83 - 1.17 \times 1.17\)[/tex], resembles the difference of squares formula. The identity for the difference of squares is:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Here, let:
[tex]\[
a = 7.83 \quad \text{and} \quad b = 1.17
\][/tex]
3. Apply the Identity:
Using the identity, we can rewrite the numerator:
[tex]\[
7.83^2 - 1.17^2 = (7.83 + 1.17)(7.83 - 1.17)
\][/tex]
4. Calculate [tex]\(a + b\)[/tex] and [tex]\(a - b\)[/tex]:
First, find the sum and the difference of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
a + b = 7.83 + 1.17 = 9.00
\][/tex]
[tex]\[
a - b = 7.83 - 1.17 = 6.66
\][/tex]
5. Substitute Back into the Expression:
Now substitute these values back into the rewritten form of the numerator:
[tex]\[
(7.83 + 1.17)(7.83 - 1.17) = 9.00 \times 6.66
\][/tex]
6. Simplify the Expression:
The original expression now becomes:
[tex]\[
\frac{(7.83 + 1.17)(7.83 - 1.17)}{6.66} = \frac{9.00 \times 6.66}{6.66}
\][/tex]
7. Cancel the Common Factor:
The common factor [tex]\(6.66\)[/tex] in the numerator and the denominator cancels out:
[tex]\[
\frac{9.00 \times 6.66}{6.66} = 9.00
\][/tex]
So, the value of the given expression is [tex]\(9.00\)[/tex].
Thus, we have:
[tex]\[
\boxed{9.00}
\][/tex]
The value of the numerator [tex]\(7.83^2 - 1.17^2\)[/tex] is [tex]\(59.94\)[/tex], and the final result after dividing by the denominator [tex]\(6.66\)[/tex] is [tex]\(9.00\)[/tex].
