May 2026

Evaluate 48³ − 30³ − 18³ Evaluate: \[ 48^3 – 30^3 – 18^3 \] Solution: Rewrite the expression: \[ 48^3 + (-30)^3 + (-18)^3 \] Since, \[ 48-30-18=0 \] Using identity: \[ a^3+b^3+c^3=3abc \quad \text{when} \quad a+b+c=0 \] Take \[ a=48,\qquad b=-30,\qquad c=-18 \] \[ 48+(-30)+(-18)=0 \] Therefore, \[ 48^3-30^3-18^3 = 3(48)(-30)(-18) \] \[ = […]

Evaluate 25³ − 75³ + 50³ Evaluate: \[ 25^3 – 75^3 + 50^3 \] Solution: Rewrite the expression: \[ 25^3 + 50^3 – 75^3 \] Since, \[ 25+50=75 \] Using identity: \[ a^3+b^3+c^3=3abc \quad \text{when} \quad a+b+c=0 \] Take \[ a=25,\qquad b=50,\qquad c=-75 \] \[ 25+50-75=0 \] Therefore, \[ 25^3+50^3-75^3 = 3(25)(50)(-75) \] \[ =

Find the Product Find the Product: \[ (3x – 4y + 5z) \] \[ (9x^2 + 16y^2 + 25z^2 + 12xy – 15zx + 20yz) \] Solution: Using identity: \[ (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc \] Rewrite: \[ 3x-4y+5z = 3x+5z-4y \] So, \[ a=3x,\qquad b=5z,\qquad c=-4y \] \[ (3x – 4y + 5z) \] \[ (9x^2

Find the Product Find the Product: \[ (2a – 3b – 2c) \] \[ (4a^2 + 9b^2 + 4c^2 + 6ab – 6bc + 4ca) \] Solution: Using identity: \[ (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc \] Rewrite: \[ 2a-3b-2c = 2a+(-3b)+(-2c) \] So, \[ a=2a,\qquad b=-3b,\qquad c=-2c \] \[ (2a – 3b – 2c) \] \[ (4a^2

Find the Product Find the Product: \[ (4x – 3y + 2z) \] \[ (16x^2 + 9y^2 + 4z^2 + 12xy + 6yz – 8zx) \] Solution: Using identity: \[ (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc \] Rewrite: \[ 4x-3y+2z = 4x+2z-3y \] So, \[ a=4x,\qquad b=2z,\qquad c=-3y \] \[ (4x – 3y + 2z) \] \[ (16x^2

Evaluate Using Identity Question: If \[ y=1 \] find the value of: \[ \left(5y + \frac{15}{y}\right) \left(25y^2 – 75 + \frac{225}{y^2}\right) \] Solution: Rearranging the terms: \[ \left(5y + \frac{15}{y}\right) \left(25y^2 – 5y\cdot\frac{15}{y} + \frac{225}{y^2}\right) \] Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=5y,\qquad b=\frac{15}{y} \] \[ = (5y)^3 + \left(\frac{15}{y}\right)^3 \] Substituting \[ y=1

Find the Product Find the Product: \[ (3x + 2y + 2z) \] \[ (9x^2 + 4y^2 + 4z^2 – 6xy – 4yz – 6zx) \] Solution: Using identity: \[ (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc \] Here, \[ a=3x,\qquad b=2y,\qquad c=2z \] \[ (3x + 2y + 2z) \] \[ (9x^2 + 4y^2 + 4z^2 – 6xy

Evaluate Using Identity Question: If \[ x=-2 \] find the value of: \[ \left(\frac{2}{x} – \frac{x}{2}\right) \left(\frac{4}{x^2} + \frac{x^2}{4} + 1\right) \] Solution: Rearranging the terms: \[ \left(\frac{2}{x} – \frac{x}{2}\right) \left(\frac{4}{x^2} + \frac{2}{x}\cdot\frac{x}{2} + \frac{x^2}{4}\right) \] Using identity: \[ (a-b)(a^2+ab+b^2)=a^3-b^3 \] Here, \[ a=\frac{2}{x},\qquad b=\frac{x}{2} \] \[ = \left(\frac{2}{x}\right)^3 – \left(\frac{x}{2}\right)^3 \] Substituting \[ x=-2

Evaluate Using Identity Question: If \[ x=-2 \quad \text{and} \quad y=1 \] find the value of: \[ (4y^2 – 9x^2)(16y^4 + 36x^2y^2 + 81x^4) \] Solution: Using identity: \[ (a-b)(a^2+ab+b^2)=a^3-b^3 \] Here, \[ a=4y^2,\qquad b=9x^2 \] \[ (4y^2 – 9x^2)(16y^4 + 36x^2y^2 + 81x^4) \] \[ = (4y^2)^3-(9x^2)^3 \] \[ = 64y^6-729x^6 \] Substituting \[

Find a³ − b³ Question: If \[ a-b=6 \quad \text{and} \quad ab=20 \] find: \[ a^3-b^3 \] Solution: Using identity: \[ a^3-b^3=(a-b)^3+3ab(a-b) \] Substituting the given values: \[ a^3-b^3=(6)^3+3(20)(6) \] \[ =216+360 \] \[ =576 \] Next Question / Full Exercise