# Find each of the following products:

Question:

Find each of the following products:

(i) (x + 3)(x − 3)

(ii) (2x + 5)(2x − 5)

(iii) (8 + x)(8 − x)

(iv) (7x + 11y)(7x − 11y)

(v) $\left(5 x^{2}+\frac{3}{4} y^{2}\right)\left(5 x^{2}-\frac{3}{4} y^{2}\right)$

(vi) $\left(\frac{4 x}{5}-\frac{5 y}{3}\right)\left(\frac{4 x}{5}+\frac{5 y}{3}\right)$

(vii) $\left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right)$

(viii) $\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right)$

Solution:

(i) We have:

$(x+3)(x-3)$

$=x^{2}-9$$\quad\left[\right. using \left.(a+b)(a-b)=a^{2}-b^{2}\right] (ii) We have: (2 x+5)(2 x-5) =4 x^{2}-25$$\quad\left[\right.$ using $\left.(a+b)(a-b)=a^{2}-b^{2}\right]$

(iii) We have:

$(8+x)(8-x)$

$=64-x^{2}$$\quad\left[\right. using \left.(a+b)(a-b)=a^{2}-b^{2}\right] (iv) We have: (7 x+11 y)(7 x-11 y) =49 x^{2}-121 y^{2}$$\quad\left[\right.$ using $\left.(a+b)(a-b)=a^{2}-b^{2}\right]$

(v) We have:

$\left(5 x^{2}+\frac{3}{4} y^{2}\right)\left(5 x^{2}-\frac{3}{4} y^{2}\right)$=25 x^{4}-\frac{9}{16} y^{4}$$\quad\left[\right. using \left.(a+b)(a-b)=a^{2}-b^{2}\right] (vi) We have: \left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right) =x^{2}-\frac{1}{x^{2}}$$\quad\left[\right.$using$\left.(a+b)(a-b)=a^{2}-b^{2}\right]$(vii) We have:$\left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right)

$=x^{2}-\frac{1}{x^{2}}$$\quad\left[\right. using \left.(a+b)(a-b)=a^{2}-b^{2}\right] (viii) We have: \left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right) =\frac{1}{x^{2}}-\frac{1}{y^{2}}$$\quad\left[\operatorname{using}(a+b)(a-b)=a^{2}-b^{2}\right]$

(ix) We have:

$\left(2 a+\frac{3}{b}\right)\left(2 a-\frac{3}{b}\right)$

$=4 a^{2}-\frac{9}{b^{2}}$$\quad\left[\right.$ using $\left.(a+b)(a-b)=a^{2}-b^{2}\right]$