Find each of the following products:
(i) $(x-4)(x-4)$
(ii) $(2 x-3 y)(2 x-3 y)$
(iii) $\left(\frac{3}{4} x-\frac{5}{6} y\right)\left(\frac{3}{4} x-\frac{5}{6} y\right)$
(iv) $\left(x-\frac{3}{x}\right)\left(x-\frac{3}{x}\right)$
(v) $\left(\frac{1}{3} x^{2}-9\right)\left(\frac{1}{3} x^{2}-9\right)$
(vi) $\left(\frac{1}{2} y^{2}-\frac{1}{3} y\right)\left(\frac{1}{2} y^{2}-\frac{1}{3} y\right)$
(i) We have:
$(x-4)(x-4)$
$=(x-4)^{2}$
$=x^{2}-2 \times x \times 4+4^{2}$ $\left[\right.$ using $\left.(a-b)^{2}=a^{2}-2 a b+b^{2}\right]$
$=x^{2}-8 x+16$
(ii) We have:
$(2 x-3 y)(2 x-3 y)$
$=(2 x-3 y)^{2}$
$=(2 x)^{2}-2 \times 2 x \times 3 y+(3 y)^{2}$ $\left[\right.$ using $\left.(a-b)^{2}=a^{2}-2 a b+b^{2}\right]$
$=4 x^{2}-12 x y+9 y^{2}$
(iii) $\left(\frac{3}{4} x-\frac{5}{6} y\right)\left(\frac{3}{4} x-\frac{5}{6} y\right)$
$=\left(\frac{3}{4} x-\frac{5}{6} y\right)^{2}$
$=\left(\frac{3}{4} x\right)^{2}-2 \times \frac{3}{4} x \times \frac{5}{6} y+\left(\frac{5}{6} y\right)^{2}$ $\left[\right.$ using $\left.(a-b)^{2}=a^{2}-2 a b+b^{2}\right]$
$=\frac{9}{16} x^{2}-\frac{15}{12} x y+\frac{25}{36} y^{2}$
(iv) We have:
$\left(x-\frac{3}{x}\right)\left(x-\frac{3}{x}\right)$
$=\left(x-\frac{3}{x}\right)^{2}$
$=(x)^{2}-2 \times x \times \frac{3}{x}+\left(\frac{3}{x}\right)^{2}$ $\left[\right.$ using $\left.(a-b)^{2}=a^{2}-2 a b+b^{2}\right]$
$=x^{2}-6+\frac{9}{x^{2}}$
(v) We have:
$\left(\frac{1}{3} x^{2}-9\right)\left(\frac{1}{3} x^{2}-9\right)$
$=\left(\frac{1}{3} x^{2}-9\right)^{2}$
$=\left(\frac{1}{3} x^{2}\right)^{2}-2 \times \frac{1}{3} x^{2} \times 9+(9)^{2}$ $\left[\operatorname{using}(a-b)^{2}=a^{2}-2 a b+b^{2}\right]$
$=\frac{1}{9} x^{4}-6 x^{2}+81$
(vi) We have:
$\left(\frac{1}{2} y^{2}-\frac{1}{3} y\right)\left(\frac{1}{2} y^{2}-\frac{1}{3} y\right)$
$=\left(\frac{1}{2} y^{2}-\frac{1}{3} y\right)^{2}$
$=\left(\frac{1}{2} y^{2}\right)^{2}-2 \times \frac{1}{2} y^{2} \times \frac{1}{3} y+\left(\frac{1}{3} y\right)^{2}$ $\left[\operatorname{using}(a-b)^{2}=a^{2}-2 a b+b^{2}\right]$
$=\frac{1}{4} y^{4}-\frac{1}{3} y^{3}+\frac{1}{9} y^{2}$
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