Educational

Find the Value Using Identity Find the Value of \(27x^3+8y^3\), if \(3x+2y=20\) and \(xy=\frac{14}{9}\) Solution: Using identity: \[ a^3+b^3=(a+b)^3-3ab(a+b) \] Here, \[ a=3x,\quad b=2y \] \[ a+b=20 \] \[ ab=(3x)(2y)=6xy =6\left(\frac{14}{9}\right) =\frac{28}{3} \] \[ 27x^3+8y^3 = (20)^3-3\left(\frac{28}{3}\right)(20) \] \[ = 8000-560 \] \[ =7440 \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ 3x+2y=14 \] \[ xy=8 \] Find: \[ 27x^3+8y^3 \] Solution: Using identity: \[ a^3+b^3=(a+b)^3-3ab(a+b) \] Here, \[ a=3x,\quad b=2y \] \[ a+b=14 \] \[ ab=(3x)(2y)=6xy=6(8)=48 \] \[ 27x^3+8y^3 = (14)^3-3(48)(14) \] \[ = 2744-2016 \] \[ =728 \] Next Question / Full Exercise

Calculate Values Using Identity Calculate the Following Values \[ x+\frac{1}{x}=3 \] Find: \[ x^2+\frac{1}{x^2},\quad x^3+\frac{1}{x^3},\quad x^4+\frac{1}{x^4} \] Solution: Using identity: \[ \left(x+\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}+2 \] \[ (3)^2 = x^2+\frac{1}{x^2}+2 \] \[ 9 = x^2+\frac{1}{x^2}+2 \] \[ x^2+\frac{1}{x^2} = 7 \] Now using identity: \[ a^3+b^3=(a+b)^3-3ab(a+b) \] \[ x^3+\frac{1}{x^3} = \left(x+\frac{1}{x}\right)^3 -3\left(x\cdot\frac{1}{x}\right)\left(x+\frac{1}{x}\right) \] \[ = (3)^3-3(1)(3)

Evaluate Using Identity Evaluate \[ 93^3-107^3 \] Solution: Using identity: \[ a^3-b^3=(a-b)(a^2+ab+b^2) \] \[ 93^3-107^3 = (93-107)(93^2+93\times107+107^2) \] \[ = (-14)(8649+9951+11449) \] \[ = (-14)(30049) \] \[ =-420686 \] Next Question / Full Exercise

Evaluate Using Identity Evaluate \[ 104^3+96^3 \] Solution: Using identity: \[ a^3+b^3=(a+b)(a^2-ab+b^2) \] \[ 104^3+96^3 = (104+96)(104^2-104\times96+96^2) \] \[ = 200(10816-9984+9216) \] \[ = 200(10048) \] \[ =2009600 \] Next Question / Full Exercise

Evaluate Using Identity Evaluate \[ 46^3+34^3 \] Solution: Using identity: \[ a^3+b^3=(a+b)(a^2-ab+b^2) \] \[ 46^3+34^3 = (46+34)(46^2-46\times34+34^2) \] \[ = 80(2116-1564+1156) \] \[ = 80(1708) \] \[ =136640 \] Next Question / Full Exercise

Evaluate Using Identity Evaluate \[ 111^3-89^3 \] Solution: Using identity: \[ a^3-b^3=(a-b)(a^2+ab+b^2) \] \[ 111^3-89^3 = (111-89)(111^2+111\times89+89^2) \] \[ = 22(12321+9879+7921) \] \[ = 22(30121) \] \[ =662662 \] Next Question / Full Exercise

Evaluate Using Identity Evaluate \[ (99)^3 \] Solution: Using identity: \[ (a-b)^3 = a^3-b^3-3ab(a-b) \] \[ (99)^3 = (100-1)^3 \] \[ = (100)^3-(1)^3-3(100)(1)(100-1) \] \[ = 1000000-1-300(99) \] \[ = 1000000-1-29700 \] \[ = 970299 \] Next Question / Full Exercise

Evaluate Using Identity Evaluate \[ (598)^3 \] Solution: Using identity: \[ (a-b)^3 = a^3-b^3-3ab(a-b) \] \[ (598)^3 = (600-2)^3 \] \[ = (600)^3-(2)^3-3(600)(2)(600-2) \] \[ = 216000000-8-3600(598) \] \[ = 216000000-8-2152800 \] \[ = 213847192 \] Next Question / Full Exercise

Evaluate Using Identity Evaluate \[ (10.4)^3 \] Solution: Using identity: \[ (a+b)^3 = a^3+b^3+3ab(a+b) \] \[ (10.4)^3 = (10+0.4)^3 \] \[ = (10)^3+(0.4)^3+3(10)(0.4)(10+0.4) \] \[ = 1000+0.064+12(10.4) \] \[ = 1000+0.064+124.8 \] \[ = 1124.864 \] Next Question / Full Exercise