GMAT Exponents, Roots and Quadratic Equations

Exponents, roots and quadratic equations form the algebra core of GMAT Quantitative Reasoning. GMAC sets the section at 21 Problem Solving questions in 45 minutes, with no calculator allowed. These rules recur across almost every other Quant topic, from ratios to sequences, so the time you spend here pays off across the whole section.

GMAC frames Quant as a reasoning test, not a computation test. Official guidance states the section requires only elementary algebra and arithmetic knowledge, and rewards analytical reasoning over raw calculation. This guide covers the exponent and root rules, quadratic factoring methods, function notation format, and the sign traps GMAC builds into these questions, with original worked examples for each.

How GMAC Defines These Content Areas

GMAC's official skill groups sort Quant questions into named content areas instead of loose topics. The table below defines the four areas covered in this guide.

Roots do not get a named skill group of their own. Radical manipulation sits inside Algebraic Manipulation, and root-of-exponent problems sit inside Exponents, per the same skill groups breakdown. For first-degree equations and inequalities, see our guide to GMAT linear equations and inequalities. For the full Quant section format, see our GMAT Quant Reasoning guide.

Exponent Rules and Simplification

Core exponent rules

Memorize six rules and you handle nearly every exponent simplification GMAT Quant asks for.

• Product rule: x^a times x^b equals x^(a+b)
• Quotient rule: x^a divided by x^b equals x^(a-b)
• Power of a power: (x^a)^b equals x^(ab)
• Negative exponent: x^(-a) equals 1 divided by x^a
• Zero exponent: x^0 equals 1, for any nonzero x
• Fractional exponent: x^(1/n) equals the nth root of x

Root rules and simplification

Reach for these radical rules whenever a question mixes roots with exponents, and simplify before you compute.

• Product rule: the square root of a times the square root of b equals the square root of ab, for a and b at or above zero
• Quotient rule: the square root of a divided by the square root of b equals the square root of a divided by b, for a at or above zero and b above zero
• Rationalizing a denominator: multiply the top and bottom of a fraction by the radical to remove it from the denominator
• Radical-exponent form: the nth root of x^m equals x^(m/n)

Factoring and Solving Quadratic Equations

GMAC's Quadratic/Other Equalities/Inequalities content area covers any equation or inequality above the first degree, including inequalities like (x-2)(x+5) > 0 and mixed exponential equations like 2^(x-1) = 8^(3x). The name is broader than it sounds, so treat every non-linear equation you meet in Quant as part of this group. You solve all of it without a calculator, so pick the route with the least arithmetic. Factoring is fastest when the coefficients are small, and the quadratic formula is your fallback when they are not.

Common factoring patterns

Recognize these three patterns to factor faster under a strict clock.

• Difference of squares: a^2 - b^2 equals (a+b)(a-b)
• Perfect square trinomial: a^2 + 2ab + b^2 equals (a+b)^2
• Simple trinomial: x^2 + (p+q)x + pq equals (x+p)(x+q)

Solving methods

Three methods solve most GMAT quadratics, so pick by how clean the numbers look.

1. Factor the expression and set each factor equal to zero
2. Complete the square when the expression does not factor cleanly
3. Apply the quadratic formula: x equals negative b plus or minus the square root of (b squared minus 4ac), divided by 2a

Function Notation Questions

GMAC calls this content area Newly Defined Functions. These problems center on a function defined inside the question itself, either a real function outside standard GMAT background, such as the greatest-integer function, or a fully made-up symbol. Read the definition twice before you substitute, since the whole question turns on its exact wording. Treat the symbol as a set of instructions, not as a formula you are expected to know already. Substitute one value at a time and write each step down.

For example, define an operation a $ b to mean a^2 minus b. Evaluate 3 $ 2 by substituting directly to get 3^2 minus 2, which equals 7. Newly defined functions never require outside knowledge. The definition GMAC gives you inside the question contains everything needed to solve it.

Sign and Extraneous-Root Traps

Track signs carefully and you avoid most careless errors on exponent, root and quadratic questions. These five traps cause the bulk of lost points.

• Squaring both sides of an equation introduces extraneous roots. Check every solution against the original equation before you select an answer.
• A square root symbol returns only the nonnegative root by definition. The square root of 9 equals 3, not negative 3, even though x^2 = 9 has two solutions, 3 and negative 3.
• A negative exponent flips the base into a fraction. It does not make the value negative: 2^(-3) equals 1/8, a positive number.
• Multiplying or dividing an inequality by a negative number flips the inequality sign.
• An absolute value equation splits into two cases, one positive and one negative. Check both against the original equation for extraneous solutions.

Sample Problem Walkthroughs

Work these three original problems yourself, then compare your steps against the solutions below. Each one applies a rule from the sections above, so treat any gap as a signal to re-read the rule. Keep your written work short, since the section clock leaves little room for rewriting.

Exponent equation

Solve for x: 4^(x+1) equals 8^(x-1). Rewrite both sides with base 2: 4 equals 2^2 and 8 equals 2^3, so the equation becomes 2^(2(x+1)) equals 2^(3(x-1)). Matching exponents gives 2x + 2 equals 3x minus 3. Solving for x gives x equals 5.

Extraneous root

Solve for x: the square root of (x+3) equals x minus 3. Square both sides: x + 3 equals x^2 minus 6x + 9, which rearranges to x^2 minus 7x + 6 equals 0. Factoring gives (x-1)(x-6) equals 0, so x equals 1 or x equals 6. Test both candidates in the original equation before you answer.

Checking x equals 1 gives the square root of 4, which equals 2, against a right side of 1 minus 3, which equals negative 2. The two sides do not match, so x equals 1 is extraneous. Checking x equals 6 gives the square root of 9, which equals 3, against a right side of 6 minus 3, which also equals 3. The equation holds, so x equals 6 is your only valid solution.

Function notation

Define f(n) equals n^2 minus n for positive integers n. Find f(4) minus f(2). f(4) equals 16 minus 4, or 12. f(2) equals 4 minus 2, or 2. The difference, 12 minus 2, equals 10.

Calculator and Scratch Paper Rules for Quant

You get no calculator, physical or on-screen, during Quantitative Reasoning, according to GMAC's calculator and scratch paper policy. Verbal Reasoning also runs without a calculator. Data Insights is the only section with a built-in on-screen calculator. Personal calculators are not permitted unless approved through GMAC Accommodations.

Note-taking works differently by test format. Test-center candidates do not bring their own paper or pens. GMAC provides a note-taking booklet and marker, which stay in the testing room. Online candidates use an erasable whiteboard meeting GMAT requirements instead of physical paper.

Where to Practice These Skills

Practice each skill area on its own before you mix them under time pressure. We offer free Powers & Roots and Quadratics & Functions drills at three difficulty levels, untimed with instant scoring and no signup required. Visit our GMAT prep page for the full mock and section-practice library, and check your target score with our GMAT score calculator.

Sources

• GMAT Exam Content - mba.com
• GMAT Quantitative Skill Groups - mba.com
• GMAT Calculator and Scratch Paper/Note-Taking Policies - mba.com Support