Find the Value Using Identity Find the Value \[ x+\frac{1}{x}=5 \] Find: \[ x^3+\frac{1}{x^3} \] Solution: Using identity: \[ a^3+b^3=(a+b)^3-3ab(a+b) \] Here, \[ a=x,\quad b=\frac{1}{x} \] \[ ab=x\left(\frac{1}{x}\right)=1 \] \[ x^3+\frac{1}{x^3} = \left(x+\frac{1}{x}\right)^3 -3\left(x\cdot\frac{1}{x}\right)\left(x+\frac{1}{x}\right) \] \[ = (5)^3-3(1)(5) \] \[ = 125-15 \] \[ =110 \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ a-b=4 \] \[ ab=21 \] Find: \[ a^3-b^3 \] Solution: Using identity: \[ a^3-b^3=(a-b)^3+3ab(a-b) \] \[ a^3-b^3 = (4)^3+3(21)(4) \] \[ = 64+252 \] \[ =316 \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ a+b=10 \] \[ ab=21 \] Find: \[ a^3+b^3 \] Solution: Using identity: \[ a^3+b^3=(a+b)^3-3ab(a+b) \] \[ a^3+b^3 = (10)^3-3(21)(10) \] \[ = 1000-630 \] \[ =370 \] Next Question / Full Exercise

Cube of Binomial Expression Find the Cube of the Following Binomial Expression \[ 4-\frac{1}{3x} \] Solution: Using identity: \[ (a-b)^3 = a^3-b^3-3ab(a-b) \] \[ \left(4-\frac{1}{3x}\right)^3 \] \[ = (4)^3 – \left(\frac{1}{3x}\right)^3 – 3(4)\left(\frac{1}{3x}\right) \left(4-\frac{1}{3x}\right) \] \[ = 64 – \frac{1}{27x^3} – \frac{4}{x} \left(4-\frac{1}{3x}\right) \] \[ = 64 – \frac{1}{27x^3} – \frac{16}{x} + \frac{4}{3x^2} \] Next

Cube of Binomial Expression Find the Cube of the Following Binomial Expression \[ 2x+\frac{3}{x} \] Solution: Using identity: \[ (a+b)^3 = a^3+b^3+3ab(a+b) \] \[ \left(2x+\frac{3}{x}\right)^3 \] \[ = (2x)^3 + \left(\frac{3}{x}\right)^3 + 3(2x)\left(\frac{3}{x}\right) \left(2x+\frac{3}{x}\right) \] \[ = 8x^3 + \frac{27}{x^3} + 18\left(2x+\frac{3}{x}\right) \] \[ = 8x^3 + \frac{27}{x^3} + 36x + \frac{54}{x} \] Next Question

Cube of Binomial Expression Find the Cube of the Following Binomial Expression \[ \frac{3}{x}-\frac{2}{x^2} \] Solution: Using identity: \[ (a-b)^3 = a^3-b^3-3ab(a-b) \] \[ \left(\frac{3}{x}-\frac{2}{x^2}\right)^3 \] \[ = \left(\frac{3}{x}\right)^3 – \left(\frac{2}{x^2}\right)^3 – 3\left(\frac{3}{x}\right)\left(\frac{2}{x^2}\right) \left(\frac{3}{x}-\frac{2}{x^2}\right) \] \[ = \frac{27}{x^3} – \frac{8}{x^6} – \frac{18}{x^3} \left(\frac{3}{x}-\frac{2}{x^2}\right) \] \[ = \frac{27}{x^3} – \frac{8}{x^6} – \frac{54}{x^4} + \frac{36}{x^5} \] Next

Simplify Expression Using Identity Simplify the Following Expression \[ (x^2-x+1)^2-(x^2+x+1)^2 \] Solution: Using identity: \[ a^2-b^2=(a-b)(a+b) \] \[ = \left[(x^2-x+1)-(x^2+x+1)\right] \left[(x^2-x+1)+(x^2+x+1)\right] \] \[ = (-2x)(2x^2+2) \] \[ = -4x(x^2+1) \] Next Question / Full Exercise

Cube of Binomial Expression Find the Cube of the Following Binomial Expression \[ \frac{1}{x}+\frac{y}{3} \] Solution: Using identity: \[ (a+b)^3 = a^3+b^3+3ab(a+b) \] \[ \left(\frac{1}{x}+\frac{y}{3}\right)^3 \] \[ = \left(\frac{1}{x}\right)^3 + \left(\frac{y}{3}\right)^3 + 3\left(\frac{1}{x}\right)\left(\frac{y}{3}\right) \left(\frac{1}{x}+\frac{y}{3}\right) \] \[ = \frac{1}{x^3} + \frac{y^3}{27} + \frac{y}{x} \left(\frac{1}{x}+\frac{y}{3}\right) \] \[ = \frac{1}{x^3} + \frac{y^3}{27} + \frac{y}{x^2} + \frac{y^2}{3x} \] Next

Simplify Expression Using Identity Simplify the Following Expression \[ (x+y-2z)^2-x^2-y^2-3z^2+4xy \] Solution: Using identity: \[ (a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \] \[ (x+y-2z)^2 = x^2+y^2+(2z)^2+2xy-2(y)(2z)-2(x)(2z) \] \[ = x^2+y^2+4z^2+2xy-4yz-4xz \] \[ (x+y-2z)^2-x^2-y^2-3z^2+4xy \] \[ = x^2+y^2+4z^2+2xy-4yz-4xz -x^2-y^2-3z^2+4xy \] \[ = z^2+6xy-4yz-4xz \] Next Question / Full Exercise

Simplify Expression Using Identity Simplify the Following Expression \[ (x+y+z)^2+\left(x+\frac{y}{2}+\frac{z}{3}\right)^2-\left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^2 \] Solution: Using identity: \[ (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca \] \[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \] \[ \left(x+\frac{y}{2}+\frac{z}{3}\right)^2 = x^2+\frac{y^2}{4}+\frac{z^2}{9} +xy+\frac{yz}{3}+\frac{2xz}{3} \] \[ \left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^2 = \frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16} +\frac{xy}{3}+\frac{yz}{6}+\frac{xz}{4} \] \[ (x+y+z)^2+\left(x+\frac{y}{2}+\frac{z}{3}\right)^2-\left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^2 \] \[ = \frac{7x^2}{4} +\frac{41y^2}{36} +\frac{127z^2}{144} +\frac{8xy}{3} +\frac{13yz}{6} +\frac{29xz}{12} \] Next Question / Full Exercise