Certainly! Let's find the products of the given expressions step-by-step.
### Part (a)
Given the expression [tex]\((8x - 5y)(8x + 5y)\)[/tex]:
This is a difference of squares, which follows the formula [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex].
Here, [tex]\(a = 8x\)[/tex] and [tex]\(b = 5y\)[/tex].
Using the formula:
[tex]\[
(8x - 5y)(8x + 5y) = (8x)^2 - (5y)^2
\][/tex]
Calculate the squares:
[tex]\[
(8x)^2 = 64x^2
\][/tex]
[tex]\[
(5y)^2 = 25y^2
\][/tex]
So:
[tex]\[
(8x - 5y)(8x + 5y) = 64x^2 - 25y^2
\][/tex]
### Part (b)
Given the expression [tex]\((9a^2 + b^2)(9a^2 - b^2)\)[/tex]:
This is another example of a difference of squares.
Here, [tex]\(a = 9a^2\)[/tex] and [tex]\(b = b^2\)[/tex].
Using the formula:
[tex]\[
(9a^2 + b^2)(9a^2 - b^2) = (9a^2)^2 - (b^2)^2
\][/tex]
Calculate the squares:
[tex]\[
(9a^2)^2 = 81a^4
\][/tex]
[tex]\[
(b^2)^2 = b^4
\][/tex]
So:
[tex]\[
(9a^2 + b^2)(9a^2 - b^2) = 81a^4 - b^4
\][/tex]
### Part (c)
Given the expression [tex]\((3.2p - 2.4q)(3.2p + 2.4q)\)[/tex]:
This is also a difference of squares.
Here, [tex]\(a = 3.2p\)[/tex] and [tex]\(b = 2.4q\)[/tex].
Using the formula:
[tex]\[
(3.2p - 2.4q)(3.2p + 2.4q) = (3.2p)^2 - (2.4q)^2
\][/tex]
Calculate the squares:
[tex]\[
(3.2p)^2 = (3.2)^2 (p)^2 = 10.24p^2
\][/tex]
[tex]\[
(2.4q)^2 = (2.4)^2 (q)^2 = 5.76q^2
\][/tex]
So:
[tex]\[
(3.2p - 2.4q)(3.2p + 2.4q) = 10.24p^2 - 5.76q^2
\][/tex]
### Part (d)
Given the expression [tex]\((6x - 7)(6x + 7)\)[/tex]:
This is another difference of squares.
Here, [tex]\(a = 6x\)[/tex] and [tex]\(b = 7\)[/tex].
Using the formula:
[tex]\[
(6x - 7)(6x + 7) = (6x)^2 - 7^2
\][/tex]
Calculate the squares:
[tex]\[
(6x)^2 = 36x^2
\][/tex]
[tex]\[
7^2 = 49
\][/tex]
So:
[tex]\[
(6x - 7)(6x + 7) = 36x^2 - 49
\][/tex]
### Summary
The products of the given expressions are:
- [tex]\((8x - 5y)(8x + 5y) = 64x^2 - 25y^2\)[/tex]
- [tex]\((9a^2 + b^2)(9a^2 - b^2) = 81a^4 - b^4\)[/tex]
- [tex]\((3.2p - 2.4q)(3.2p + 2.4q) = 10.24p^2 - 5.76q^2\)[/tex]
- [tex]\((6x - 7)(6x + 7) = 36x^2 - 49\)[/tex]
