GMAT Exponents & Applied Algebra: A Guide to Calculation & Shortcuts

This is the second part of our guide to GMAT Number Properties. Here, we focus on the algebraic tools and computational shortcuts that are essential for solving complex problems with speed and accuracy.

Section 1: Fundamentals of Exponents (Indices)

The Foundation: What is an Exponent?

Start with the core definitions, including positive, negative, zero, and fractional exponents. Build an intuitive understanding with our interactive explorer.

Repeated Multiplication

$$a^n$$ means multiplying 'a' by itself 'n' times.

Core Application: Properties & Laws of Exponents

Master the essential rules for manipulating exponents, such as the product, quotient, and power rules, and understand the critical 'tower of powers' distinction.

Basic Operations

$$ a^m \times a^n $$ = $$ a^{m+n} $$

Verify with numbers:

$$ a^m / a^n $$ = $$ a^{m-n} $$

Verify with numbers:

Section 2: Essential Formulas & Calculation Shortcuts

Key Skill: Essential Algebraic Identities

Memorize and apply the 7 essential formulae for squares and cubes, like (a+b)² and a²-b², which are frequently tested on the GMAT.

Basic Formulae for Squares and Cubes

$$ (a+b)^2 $$ = $$ a^2 + b^2 + 2ab $$

Verify with numbers:

$$ (a-b)^2 $$ = $$ a^2 + b^2 - 2ab $$

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$$ a^2 - b^2 $$ = $$ (a+b)(a-b) $$

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$$ (a+b)^3 $$ = $$ a^3 + b^3 + 3ab(a+b) $$

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$$ (a-b)^3 $$ = $$ a^3 - b^3 - 3ab(a-b) $$

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$$ a^3 + b^3 $$ = $$ (a+b)(a^2 - ab + b^2) $$

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$$ a^3 - b^3 $$ = $$ (a-b)(a^2 + ab + b^2) $$

Verify with numbers:

Speed Trick: Squaring Shortcuts

Learn powerful mental math techniques to square numbers quickly and accurately, a critical skill for managing time on the GMAT Quant section.

Mental Math Squaring

Numbers ending in 5

Square the tens digit (n) by (n+1), then append 25.

Advanced Concept: Series Summation

or high-level questions, learn the formulas for the sum of the first 'n' numbers, their squares, and their cubes.

Summation Formulas

Sum of first n integers = $$ \frac{n(n+1)}{2} $$

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Sum of first n squares = $$ \frac{n(n+1)(2n+1)}{6} $$

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Sum of first n cubes = $$ \left[ \frac{n(n+1)}{2} \right]^2 $$

Verify with numbers:

Interactive Practice

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Expert Strategy & Insights

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The 'Symmetric Sequence' Trick for Standard Deviation

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Mastering Combinations: The Foundational Listing Method

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