To solve the problems [tex]\(27^2-26^2\)[/tex] and [tex]\(118^2-117^2\)[/tex] without actually calculating the squares, we can use the difference of squares formula. The difference of squares formula states that:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
### Part (i) [tex]\(27^2 - 26^2\)[/tex]
Let's denote [tex]\(a = 27\)[/tex] and [tex]\(b = 26\)[/tex].
Using the formula:
[tex]\[
27^2 - 26^2 = (27 - 26)(27 + 26)
\][/tex]
Now, we calculate the values inside the parentheses:
[tex]\[
27 - 26 = 1
\][/tex]
[tex]\[
27 + 26 = 53
\][/tex]
Therefore:
[tex]\[
27^2 - 26^2 = 1 \times 53 = 53
\][/tex]
So, the value of [tex]\(27^2 - 26^2\)[/tex] is [tex]\(53\)[/tex].
### Part (ii) [tex]\(118^2 - 117^2\)[/tex]
Let's denote [tex]\(a = 118\)[/tex] and [tex]\(b = 117\)[/tex].
Using the formula:
[tex]\[
118^2 - 117^2 = (118 - 117)(118 + 117)
\][/tex]
Now, we calculate the values inside the parentheses:
[tex]\[
118 - 117 = 1
\][/tex]
[tex]\[
118 + 117 = 235
\][/tex]
Therefore:
[tex]\[
118^2 - 117^2 = 1 \times 235 = 235
\][/tex]
So, the value of [tex]\(118^2 - 117^2\)[/tex] is [tex]\(235\)[/tex].
In summary:
(i) [tex]\(27^2 - 26^2 = 53\)[/tex]
(ii) [tex]\(118^2 - 117^2 = 235\)[/tex]
