Educational

Evaluate Using Identity Evaluate \[ (98)^3 \] Solution: Using identity: \[ (a-b)^3 = a^3-b^3-3ab(a-b) \] \[ (98)^3 = (100-2)^3 \] \[ = (100)^3-(2)^3-3(100)(2)(100-2) \] \[ = 1000000-8-600(98) \] \[ = 1000000-8-58800 \] \[ = 941192 \]

Evaluate Using Identity Evaluate \[ (103)^3 \] Solution: Using identity: \[ (a+b)^3 = a^3+b^3+3ab(a+b) \] \[ (103)^3 = (100+3)^3 \] \[ = (100)^3+(3)^3+3(100)(3)(100+3) \] \[ = 1000000+27+900(103) \] \[ = 1000000+27+92700 \] \[ = 1092727 \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ 3x-2y=11 \] \[ xy=12 \] Find: \[ 27x^3-8y^3 \] Solution: Using identity: \[ a^3-b^3=(a-b)^3+3ab(a-b) \] Here, \[ a=3x,\quad b=2y \] \[ a-b=11 \] \[ ab=(3x)(2y)=6xy=6(12)=72 \] \[ 27x^3-8y^3 = (11)^3+3(72)(11) \] \[ = 1331+2376 \] \[ =3707 \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ x-\frac{1}{x}=3+2\sqrt{2} \] Find: \[ x^3-\frac{1}{x^3} \] Solution: Using identity: \[ a^3-b^3=(a-b)^3+3ab(a-b) \] Here, \[ a=x,\quad b=\frac{1}{x},\quad ab=1 \] \[ x^3-\frac{1}{x^3} = \left(x-\frac{1}{x}\right)^3 +3\left(x-\frac{1}{x}\right) \] \[ = (3+2\sqrt{2})^3 +3(3+2\sqrt{2}) \] \[ = (99+70\sqrt{2}) +(9+6\sqrt{2}) \] \[ = 108+76\sqrt{2} \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ x^2+\frac{1}{x^2}=98 \] Find: \[ x^3+\frac{1}{x^3} \] Solution: Using identity: \[ \left(x+\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}+2 \] \[ \left(x+\frac{1}{x}\right)^2 = 98+2 \] \[ \left(x+\frac{1}{x}\right)^2 = 100 \] \[ x+\frac{1}{x} = 10 \] Now using identity: \[ a^3+b^3=(a+b)^3-3ab(a+b) \] Here, \[ a=x,\quad b=\frac{1}{x},\quad ab=1 \] \[ x^3+\frac{1}{x^3} = \left(x+\frac{1}{x}\right)^3 -3\left(x+\frac{1}{x}\right)

Find the Value Using Identity Find the Value \[ x^2+\frac{1}{x^2}=51 \] Find: \[ x^3-\frac{1}{x^3} \] Solution: Using identity: \[ \left(x-\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}-2 \] \[ \left(x-\frac{1}{x}\right)^2 = 51-2 \] \[ \left(x-\frac{1}{x}\right)^2 = 49 \] \[ x-\frac{1}{x} = 7 \] Now using identity: \[ a^3-b^3=(a-b)^3+3ab(a-b) \] Here, \[ a=x,\quad b=\frac{1}{x},\quad ab=1 \] \[ x^3-\frac{1}{x^3} = \left(x-\frac{1}{x}\right)^3 +3\left(x-\frac{1}{x}\right)

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, \[ a=x,\quad b=\frac{1}{x} \] \[ ab=x\left(\frac{1}{x}\right)=1 \] \[ x^3-\frac{1}{x^3} = \left(x-\frac{1}{x}\right)^3 +3\left(x\cdot\frac{1}{x}\right)\left(x-\frac{1}{x}\right) \] \[ = (5)^3+3(1)(5) \] \[ = 125+15 \] \[ =140 \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ x-\frac{1}{x}=7 \] Find: \[ x^3-\frac{1}{x^3} \] Solution: Using identity: \[ a^3-b^3=(a-b)^3+3ab(a-b) \] Here, \[ a=x,\quad b=\frac{1}{x} \] \[ ab=x\left(\frac{1}{x}\right)=1 \] \[ x^3-\frac{1}{x^3} = \left(x-\frac{1}{x}\right)^3 +3\left(x\cdot\frac{1}{x}\right)\left(x-\frac{1}{x}\right) \] \[ = (7)^3+3(1)(7) \] \[ = 343+21 \] \[ =364 \] Next Question / Full Exercise

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