Algebraic Identities Practice Questions with Detailed Solutions – Level 2

Algebraic Identities Level 2 Practice Quiz: Higher Powers and Cubes

This set helps you check whether you can pick the right identity when a question is disguised. Algebraic Identities Level 2 on Aptitude360 covers higher powers of reciprocal expressions, hidden cube expansions, recovering sums from squares and three-variable identities. Every question has a worked solution, so you can trace where a step went wrong.

What the Level 2 algebraic identities questions test

• Fourth and sixth powers of a variable plus its reciprocal
• Recovering a sum or product from given squares, including a sign condition
• Recognising a full cube expansion written out term by term
• Difference of cubes, including cubes of expressions with coefficients
• Sum of three cubes using the sum, pairwise products and product of three values
• Sum of squared differences of three variables
• Values of irrational numbers and their reciprocals raised to a power
• Using a quadratic equation to get a reciprocal expression

Key formulas for Level 2 identity questions

• Sum of cubes: \( a^{3}+b^{3}=(a+b)^{3}-3ab(a+b) \)
• Difference cube: \( (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} \)
• Reciprocal cube: \( x^{3}+\dfrac{1}{x^{3}}=t^{3}-3t \), \( t=x+\dfrac{1}{x} \)
• Fourth power: \( x^{4}+\dfrac{1}{x^{4}}=\left(x^{2}+\dfrac{1}{x^{2}}\right)^{2}-2 \)
• Squared differences: \( (a-b)^{2}+(b-c)^{2}+(c-a)^{2}=2(a^{2}+b^{2}+c^{2}-ab-bc-ca) \)

Preparation tips for tougher identity problems

• Climb powers step by step: get the square first, then the fourth power from it.
• If a long expression has binomial-style coefficients, rewrite it as a single cube before calculating.
• For a quadratic of the form x squared plus bx plus one equal to zero, divide by x, which is nonzero, to get x plus one over x equal to minus b.
• When a square root appears, use the sign condition given in the question to pick the correct root.

Who this Level 2 set suits

It suits aspirants who have finished the basic identities and want medium-level algebra practice for quantitative aptitude sections.