Question 1: Find the following products;

1. $\left(y+9\right)\left(y-9\right)=y^2-81$
2. $\left(4+b\right)\left(4-b\right)=16-b^2$
3. $\left(3x-5\right)\left(3x+5\right)=9x^2-25$
4. $\left(a-\frac{2}{3}\right)\left(a+\frac{2}{3}\right)=a^2-\frac{4}{9}$

$\\$

Question 2:  Find the following products;

1. $\left(3x-5\right)\left(3x+5\right)=9x^2-25$
2. $\left(2+7x\right)\left(2-7x\right)=4-49x^2$
3. $\left(\frac{a}{2}+3\right)\left(\frac{a}{2}-3\right)=\frac{a^2}{4}-9$
4. $\left(4x+3y\right)\left(4x-3y\right)=16x^2-9y^2$

$\\$

Question 3:  Find the following products;

1. $\left(\frac{a}{3}-\frac{b}{4}\right)\left(\frac{a}{3}+\frac{b}{4}\right)=\frac{a^2}{9}-\frac{b^2}{16}$
2. $\left(\frac{t}{2}-\frac{1}{3}\right)\left(\frac{t}{2}+\frac{1}{3}\right)=\frac{t^2}{4}-\frac{1}{9}$

$\\$

Question 4:  Find the following products;

1. $\left(\frac{2}{x}+\frac{3}{y}\right)\left(\frac{2}{x}-\frac{3}{y}\right)=\frac{4}{x^2}-\frac{9}{y^2}$
2. $\left(\frac{1}{a}-\frac{1}{b}\right)\left(\frac{1}{a}+\frac{1}{b}\right)=\frac{1}{a^2}-\frac{1}{b^2}$
3. $\left(\frac{1}{3x}+\frac{2}{5y}\right)\left(\frac{1}{3x}-\frac{2}{5y}\right)=\frac{1}{9x^2}-\frac{2}{25y^2}$
4. $\left(1.1x-0.3y\right)\left(1.1x+0.3y\right)=1.21x^2-0.09y^2$

$\\$

Question 5:  Find the following products;

1. $\left(a^2+2b^2\right)\left(a^2-2b^2\right)=a^4-4b^4$
2. $\left(6x^2-7y^2\right)\left(6x^2+7y^2\right)=36x^4-49y^4$
3. $\left(4x^2+2yz\right)\left(2x^2-yz\right)=8x^4+4x^2yz-4x^2yz-2y^2z^2=\ 8x^4-2y^2z^2$
4. $\left(ab-\frac{3}{2}cd\right)\left(2ab+3cd\right)=2a^2b^2-3abcd+3abcd-\frac{9}{2}c^2d^2=2a^2b^2-\frac{9}{2}c^2d^2$

$\\$

Question 6:  Find the following products;

$\left(2x+3\right)\left(2x-3\right)\left(4x^2+9\right)$

$=\ \left(4x^2-9\right)\left(4x^2+9\right)$

$=\ 16x^2-81$

$\\$

$\left(x+2y\right)\left(x-2y\right)\left(x^2+4y^2\right)$

$=\ \left(x^2-4y^2\right)\left(x^2+4y^2\right)$

$=\ x^4-16y^4$

$\\$

$\left(a+bc\right)\left(a-bc\right)\left(a^2+b^2c^2\right)$

$=\ \left(a^2-b^2c^2\right)\left(a^2+b^2c^2\right)$

$={\ a}^4-b^4c^4$

$\\$

$\left(\frac{2}{5}+x\right)\left(\frac{2}{5}-x\right)\left(\frac{4}{25}+x^2\right)$

$=\ \left(\frac{4}{25}-x^2\right)\left(\frac{4}{25}+x^2\right)$

$=\ \frac{16}{625}-x^4$

$\\$

Question 7:  Using the identity $\left(a+b\right)\left(a-b\right)=\left(a^2-b^2\right)$, evaluate the following.

1. $88\times{}112=\left(100-12\right)\left(100+12\right)=1000-144=9856$
2. $153\times{}167=\left(160-7\right)\left(160+7\right)=25600-49=25551$
3. $10.8\times{}9.2=\left(10+0.8\right)\left(10-0.8\right)={10}^2-{0.8}^2=100-0.64=99.36$
4. $3\frac{1}{3}\times{}4\frac{2}{3}=\left(4-\frac{2}{3}\right)\left(4+\frac{2}{3}\right)=16-\frac{4}{9}=15\frac{5}{9}$
5. $9\frac{1}{4}\times{}15\frac{3}{4}=\left(\frac{25}{2}-3\frac{1}{4}\right)\left(\frac{25}{2}+3\frac{1}{4}\right)=145.6875$