Educational

Find the Product (2/x + 3x)(4/x² + 9x² − 6) Find the Product: \[ \left(\frac{2}{x} + 3x\right) \left(\frac{4}{x^2} + 9x^2 – 6\right) \] Solution: Rearranging the terms: \[ \left(\frac{2}{x} + 3x\right) \left(\frac{4}{x^2} – \frac{6}{1} + 9x^2\right) \] Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=\frac{2}{x},\qquad b=3x \] \[ \left(\frac{2}{x} + 3x\right) \left[\left(\frac{2}{x}\right)^2 – \left(\frac{2}{x}\right)(3x) + […]

Find the Product (3 + 5/x)(9 − 15/x + 25/x²) Find the Product: \[ \left(3 + \frac{5}{x}\right) \left(9 – \frac{15}{x} + \frac{25}{x^2}\right) \] Solution: Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=3,\qquad b=\frac{5}{x} \] \[ \left(3 + \frac{5}{x}\right) \left(9 – \frac{15}{x} + \frac{25}{x^2}\right) \] \[ = (3)^3 + \left(\frac{5}{x}\right)^3 \] \[ = 27 + \frac{125}{x^3}

Find the Product (3/x − 5/y)(9/x² + 25/y² + 15/xy) Find the Product: \[ \left(\frac{3}{x} – \frac{5}{y}\right) \left(\frac{9}{x^2} + \frac{25}{y^2} + \frac{15}{xy}\right) \] Solution: Using identity: \[ (a-b)(a^2+ab+b^2)=a^3-b^3 \] Here, \[ a=\frac{3}{x},\qquad b=\frac{5}{y} \] \[ \left(\frac{3}{x} – \frac{5}{y}\right) \left(\frac{9}{x^2} + \frac{15}{xy} + \frac{25}{y^2}\right) \] \[ = \left(\frac{3}{x}\right)^3 – \left(\frac{5}{y}\right)^3 \] \[ = \frac{27}{x^3} – \frac{125}{y^3}

Find the Product (x/2 + 2y)(x²/4 − xy + 4y²) Find the Product: \[ \left(\frac{x}{2} + 2y\right)\left(\frac{x^2}{4} – xy + 4y^2\right) \] Solution: Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=\frac{x}{2},\qquad b=2y \] \[ \left(\frac{x}{2} + 2y\right)\left(\frac{x^2}{4} – xy + 4y^2\right) \] \[ = \left(\frac{x}{2} + 2y\right) \left[\left(\frac{x}{2}\right)^2 – \left(\frac{x}{2}\right)(2y) + (2y)^2\right] \] \[ =

Find the Product (4x − 5y)(16x² + 20xy + 25y²) Find the Product: \[ (4x – 5y)(16x^2 + 20xy + 25y^2) \] Solution: Using identity: \[ (a-b)(a^2+ab+b^2)=a^3-b^3 \] Here, \[ a=4x,\qquad b=5y \] \[ (4x – 5y)(16x^2 + 20xy + 25y^2) \] \[ = (4x – 5y)\left[(4x)^2 + (4x)(5y) + (5y)^2\right] \] \[ = (4x)^3

Find the Product (7p⁴ + q)(49p⁸ − 7p⁴q + q²) Find the Product: \[ (7p^4 + q)(49p^8 – 7p^4q + q^2) \] Solution: Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=7p^4,\qquad b=q \] \[ (7p^4 + q)(49p^8 – 7p^4q + q^2) \] \[ = (7p^4 + q)\left[(7p^4)^2 – (7p^4)(q) + q^2\right] \] \[ = (7p^4)^3

Find the Product (3x + 2y)(9x² − 6xy + 4y²) Find the Product: \[ (3x + 2y)(9x^2 – 6xy + 4y^2) \] Solution: Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=3x,\qquad b=2y \] \[ (3x + 2y)(9x^2 – 6xy + 4y^2) \] \[ = (3x + 2y)\left[(3x)^2 – (3x)(2y) + (2y)^2\right] \] \[ = (3x)^3

Find the Value Using Identity Find the Value \[ x^4+\frac{1}{x^4}=119 \] Find: \[ x^3-\frac{1}{x^3} \] Solution: Using identity: \[ \left(x^2+\frac{1}{x^2}\right)^2 = x^4+\frac{1}{x^4}+2 \] \[ \left(x^2+\frac{1}{x^2}\right)^2 = 119+2 \] \[ \left(x^2+\frac{1}{x^2}\right)^2 = 121 \] \[ x^2+\frac{1}{x^2} = 11 \] Now, \[ \left(x-\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}-2 \] \[ \left(x-\frac{1}{x}\right)^2 = 11-2 \] \[ \left(x-\frac{1}{x}\right)^2 = 9 \] \[

Find Values Using Identity Find the Following Values \[ x^4+\frac{1}{x^4}=194 \] Find: \[ x^3+\frac{1}{x^3},\quad x^2+\frac{1}{x^2},\quad x+\frac{1}{x} \] Solution: Using identity: \[ \left(x^2+\frac{1}{x^2}\right)^2 = x^4+\frac{1}{x^4}+2 \] \[ \left(x^2+\frac{1}{x^2}\right)^2 = 194+2 \] \[ \left(x^2+\frac{1}{x^2}\right)^2 = 196 \] \[ x^2+\frac{1}{x^2} = 14 \] Now, \[ \left(x+\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}+2 \] \[ \left(x+\frac{1}{x}\right)^2 = 14+2 \] \[ \left(x+\frac{1}{x}\right)^2 = 16

Simplify Using Identity Simplify the Following \[ (2x-5y)^3-(2x+5y)^3 \] Solution: Let \[ a=2x-5y,\quad b=2x+5y \] Using identity: \[ a^3-b^3=(a-b)(a^2+ab+b^2) \] \[ a-b = (2x-5y)-(2x+5y) = -10y \] \[ ab = (2x-5y)(2x+5y) = 4x^2-25y^2 \] \[ a+b = 4x \] \[ a^2+ab+b^2 = (a+b)^2-3ab \] \[ = (4x)^2-3(4x^2-25y^2) \] \[ = 16x^2-12x^2+75y^2 \] \[ = 4x^2+75y^2