Alright class, let's dive into each problem one by one and expand these expressions step-by-step.
### (i) [tex]\(\left(3 x+\frac{1}{2}\right)\left(2 x+\frac{1}{3}\right)\)[/tex]
To expand this, we'll apply the distributive property:
[tex]\[
\left(3 x + \frac{1}{2}\right)\left(2 x + \frac{1}{3}\right) = 3x \cdot 2x + 3x \cdot \frac{1}{3} + \frac{1}{2} \cdot 2x + \frac{1}{2} \cdot \frac{1}{3}
\][/tex]
[tex]\[
= 6x^2 + x + x + \frac{1}{6}
\][/tex]
Combining like terms:
[tex]\[
= 6x^2 + 2x + \frac{1}{6}
\][/tex]
### (ii) [tex]\((2 a + 0.5)(7 a - 0.3)\)[/tex]
Again, we use the distributive property:
[tex]\[
(2 a + 0.5)(7 a - 0.3) = 2a \cdot 7a + 2a \cdot (-0.3) + 0.5 \cdot 7a + 0.5 \cdot (-0.3)
\][/tex]
[tex]\[
= 14a^2 -0.6a + 3.5a - 0.15
\][/tex]
Combining like terms:
[tex]\[
= 14a^2 + 2.9a - 0.15
\][/tex]
### (iii) [tex]\((9-y)(7+y)\)[/tex]
Here, we use the difference of squares formula:
[tex]\[
(9 - y)(7 + y) = 9 \cdot 7 + 9 \cdot y - y \cdot 7 - y^2
\][/tex]
[tex]\[
= 63 + 9y - 7y - y^2
\][/tex]
Combining like terms:
[tex]\[
= 63 + 2y - y^2
\][/tex]
Rearrange to get the standard form:
[tex]\[
= -y^2 + 2y + 63
\][/tex]
### (iv) [tex]\((2-z)(15-z)\)[/tex]
Again, use the distributive property:
[tex]\[
(2 - z)(15 - z) = 2 \cdot 15 + 2 \cdot (-z) - z \cdot 15 + (-z)\cdot(-z)
\][/tex]
[tex]\[
= 30 - 2z - 15z + z^2
\][/tex]
Combining like terms:
[tex]\[
= z^2 - 17z + 30
\][/tex]
### (v) [tex]\(\left(a^2 + 5\right)\left(a^2 - 3\right)\)[/tex]
Here, we use the distributive property:
[tex]\[
(a^2 + 5)(a^2 - 3) = a^2 \cdot a^2 + a^2 \cdot (-3) + 5 \cdot a^2 + 5 \cdot (-3)
\][/tex]
[tex]\[
= a^4 - 3a^2 + 5a^2 - 15
\][/tex]
Combining like terms:
[tex]\[
= a^4 + 2a^2 - 15
\][/tex]
### (vi) [tex]\((4 - ab)(8 + ab)\)[/tex]
Here, apply the distributive property:
[tex]\[
(4 - ab)(8 + ab) = 4 \cdot 8 + 4 \cdot ab - ab \cdot 8 - ab \cdot ab
\][/tex]
[tex]\[
= 32 + 4ab - 8ab - a^2b^2
\][/tex]
Combining like terms:
[tex]\[
= 32 - 4ab - a^2b^2
\][/tex]
### (vii) [tex]\((5xy - 7)(7xy + 9)\)[/tex]
Here, we use the distributive property:
[tex]\[
(5xy - 7)(7xy + 9) = 5xy \cdot 7xy + 5xy \cdot 9 - 7 \cdot 7xy - 7 \cdot 9
\][/tex]
[tex]\[
= 35x^2y^2 + 45xy - 49xy - 63
\][/tex]
Combining like terms:
[tex]\[
= 35x^2y^2 - 4xy - 63
\][/tex]
### (viii) [tex]\(\left(3a^2 - 4b^2\right)\left(8a^2 - 3b^2\right)\)[/tex]
Here, apply the distributive property:
[tex]\[
(3a^2 - 4b^2)(8a^2 - 3b^2) = 3a^2 \cdot 8a^2 + 3a^2 \cdot (-3b^2) - 4b^2 \cdot 8a^2 - 4b^2 \cdot (-3b^2)
\][/tex]
[tex]\[
= 24a^4 - 9a^2b^2 - 32a^2b^2 + 12b^4
\][/tex]
Combining like terms:
[tex]\[
= 24a^4 - 41a^2b^2 + 12b^4
\][/tex]
So these are the expanded forms of the given expressions.
