Sure! Let's solve each of the given mathematical problems step-by-step.
For the expressions where the value is given, let's confirm them first:
1) [tex]\(3^2 + 8^2 = 9 + 64 = 73\)[/tex] (Given)
3) [tex]\(7^2 + 5^2 = 49 + 25 = 74\)[/tex] (It was given as 64, but the correct answer is 74)
4) [tex]\(9^2 - 5^2 = 81 - 25 = 56\)[/tex] (Given)
Now, let's solve the others step-by-step to confirm the results:
2) [tex]\(22^2 - 20^2\)[/tex]
Using the formula for the difference of squares:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
[tex]\[
22^2 - 20^2 = (22 + 20)(22 - 20) = 42 \times 2 = 84
\][/tex]
Hence, [tex]\(22^2 - 20^2 = 84\)[/tex].
5) [tex]\(10^2 - 8^2\)[/tex]
Using the difference of squares formula:
[tex]\[
10^2 - 8^2 = (10 + 8)(10 - 8) = 18 \times 2 = 36
\][/tex]
Hence, [tex]\(10^2 - 8^2 = 36\)[/tex].
6) [tex]\(9^2 - 8^2\)[/tex]
Using the difference of squares formula:
[tex]\[
9^2 - 8^2 = (9 + 8)(9 - 8) = 17 \times 1 = 17
\][/tex]
Hence, [tex]\(9^2 - 8^2 = 17\)[/tex].
7) [tex]\(\frac{13^2 - 5^2}{24}\)[/tex]
Solving inside the fraction first:
[tex]\[
13^2 - 5^2 = 169 - 25 = 144
\][/tex]
Then, divide by 24:
[tex]\[
\frac{144}{24} = 6
\][/tex]
Hence, [tex]\(\frac{13^2 - 5^2}{24} = 6\)[/tex].
8) [tex]\(8^2 - 7^2\)[/tex]
Using the difference of squares formula:
[tex]\[
8^2 - 7^2 = (8 + 7)(8 - 7) = 15 \times 1 = 15
\][/tex]
Hence, [tex]\(8^2 - 7^2 = 15\)[/tex].
9) [tex]\(\sqrt{5^2 - 3^2}\)[/tex]
Solving inside the square root first:
[tex]\[
5^2 - 3^2 = 25 - 9 = 16
\][/tex]
Then, take the square root:
[tex]\[
\sqrt{16} = 4
\][/tex]
Hence, [tex]\(\sqrt{5^2 - 3^2} = 4\)[/tex].
10) [tex]\(7^2 - 3^3\)[/tex]
First, calculate each term:
[tex]\[
7^2 = 49
\][/tex]
[tex]\[
3^3 = 27
\][/tex]
Then, subtract them:
[tex]\[
49 - 27 = 22
\][/tex]
Hence, [tex]\(7^2 - 3^3 = 22\)[/tex].
So, here are the values for each:
2) [tex]\(22^2 - 20^2 = 84\)[/tex]
5) [tex]\(10^2 - 8^2 = 36\)[/tex]
7) [tex]\(\frac{13^2 - 5^2}{24} = 6\)[/tex]
8) [tex]\(8^2 - 7^2 = 15\)[/tex]
9) [tex]\(\sqrt{5^2 - 3^2} = 4\)[/tex]
10) [tex]\(7^2 - 3^3 = 22\)[/tex]
