4.1 Laws of Exponents & Exponential Equations

Quick Answer: An exponent records repeated multiplication. GMAT Quantitative Reasoning (Focus Edition) is 21 Problem Solving questions in 45 minutes with no calculator, and exponents sit in Official Guide Quantitative Review 2026–2027 Math Review 3.2 Algebra. Fluency means the product rule $a^m \cdot a^n = a^{m+n}$, the quotient rule $a^m / a^n = a^{m-n}$, the power rule $(a^m)^n = a^{mn}$, the reciprocal $a^{-n} = 1/a^n$, and the root $a^{m/n} = \sqrt[n]{a^m}$. To solve $b^x = b^y$ with $b > 0$ and $b \ne 1$, equate the exponents. To add or subtract exponential terms, factor the lowest common power — never add the exponents.

Logarithms are not a standalone GMAT Quant topic. If a stem looks like a change-of-base, rewrite it as an exponent identity and stay in powers. Trigonometry is out of scope.

The Eight Exponent Laws

$a^n$ means $a$ multiplied by itself $n$ times when $n$ is a positive integer. The eight identities below extend that definition to zero, negative, and rational exponents. Every identity assumes a non-zero base unless a root forces a further restriction.

Product rule (same base)

When multiplying powers of the same non-zero base, keep the base and add the exponents:

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

• $2^5 \cdot 2^3 = 2^{8} = 256$.
• Trap: do not multiply the bases. $2^5 \cdot 2^3 \ne 4^8$, and $2^3 \cdot 3^4$ is not $6^7$.

Quotient rule (same base)

When dividing powers of the same non-zero base, keep the base and subtract the denominator exponent:

$\frac{a^m}{a^n} = a^{m-n} \quad (a \ne 0)$

• $\frac{3^7}{3^4} = 3^3 = 27$.
• A hidden plus sign appears when the denominator exponent is already negative: $\frac{a^m}{a^{-n}} = a^{m-(-n)} = a^{m+n}$.

Power of a power

When a power is itself raised to a power, multiply the exponents:

$(a^m)^n = a^{mn}$

• $(5^2)^3 = 5^6 = 15{,}625$.
• Order of operations: $(a^m)^n \ne a^{m^n}$ in general. $(2^3)^2 = 2^6 = 64$, but $2^{3^2} = 2^9 = 512$ because a tower $a^{m^n}$ is evaluated from the top down.

Power of a product

$(ab)^n = a^n b^n$

• $6^x = (2 \cdot 3)^x = 2^x 3^x$.
• $(2x^3 y^2)^4 = 2^4 (x^3)^4 (y^2)^4 = 16 x^{12} y^8$.
• Trap: $(ab)^n$ is not $a^n + b^n$. Expanding a sum of powers is a different algebra move, covered as a trap in 4.3.

Power of a quotient

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \ne 0)$

• $\left(\frac{3}{4}\right)^3 = \frac{27}{64}$.

Zero exponent

$a^0 = 1 \quad (a \ne 0)$

The quotient rule forces this: $\frac{a^n}{a^n} = a^{0} = 1$. The form $0^0$ is undefined and is not a GMAT numeric constant. Parentheses decide the sign: $(-7)^0 = 1$, but $-7^0 = -(7^0) = -1$.

Negative exponent

$a^{-n} = \frac{1}{a^n}, \qquad \frac{1}{a^{-n}} = a^n \quad (a \ne 0)$

A negative exponent moves a factor across the fraction bar; it does not by itself make a value negative.

• $2^{-4} = \frac{1}{16}$.
• $\left(\frac{2}{5}\right)^{-3} = \left(\frac{5}{2}\right)^3 = \frac{125}{8}$.

Fractional (rational) exponent

$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$

For even $n$, GMAT work stays in the real numbers, so the radicand must be non-negative. For odd $n$, a negative radicand is allowed.

On scratch paper with no calculator, take the root first, then raise to the remaining power, so the integers stay small:

• $27^{4/3} = (\sqrt[3]{27})^4 = 3^4 = 81$, which is far easier than $\sqrt[3]{27^4} = \sqrt[3]{531{,}441}$.
• $16^{-3/4} = \frac{1}{(\sqrt[4]{16})^3} = \frac{1}{2^3} = \frac{1}{8}$.

Laws of Exponents Reference Table

Negative Bases and Order of Operations

Parentheses decide whether the minus is in the base

• $(-a)^n$ includes the sign in the base. Even integer $n$ gives a positive result: $(-3)^4 = 81$. Odd integer $n$ gives a negative result: $(-3)^3 = -27$.
• $-a^n$ is negation after the power: $-3^4 = -81$ and $-3^2 = -9$, which is not $(-3)^2 = 9$.

Fractional exponents on negative bases (reals only)

GMAT Quant works in the real numbers.

• Odd roots of negatives are real: $(-8)^{1/3} = \sqrt[3]{-8} = -2$.
• Even roots of negatives are undefined in the reals: $(-4)^{1/2} = \sqrt{-4}$ is not a GMAT real value. If a path of algebra produces an even root of a negative, that path is not a live real solution.

A safe rewrite of $a^{m/n}$ on a negative $a$ is to take the odd root first when $n$ is odd: $(-8)^{2/3} = \bigl(\sqrt[3]{-8}\bigr)^2 = (-2)^2 = 4$. Do not send an even root through a negative radicand on the way.

Solving Exponential Equations by Matching Bases

When the unknown sits in an exponent, rewrite every factor as a prime power, then equate exponents.

Matching-base theorem

If $b^x = b^y$ with $b > 0$ and $b \ne 1$, then $x = y$. The excluded bases matter: $b = 1$ makes $1^x = 1$ for every $x$, and $b = 0$ is not a usable exponential base on this exam.

If $a^x = b^x$ with $a \ne b$ and $a, b > 0$, the real solution is $x = 0$, because every non-zero number to the power $0$ is $1$.

Prime-power conversions to keep on scratch paper

• Powers of $2$: $4 = 2^2$, $8 = 2^3$, $16 = 2^4$, $32 = 2^5$, $64 = 2^6$, $128 = 2^7$, $256 = 2^8$, $512 = 2^9$, $1{,}024 = 2^{10}$.
• Powers of $3$: $9 = 3^2$, $27 = 3^3$, $81 = 3^4$, $243 = 3^5$, $729 = 3^6$.
• Powers of $5$: $25 = 5^2$, $125 = 5^3$, $625 = 5^4$.
• Powers of $7$: $49 = 7^2$, $343 = 7^3$.

Protocol

1. Factor every composite integer into primes.
2. Apply $(b^m)^n = b^{mn}$ until each side is a single power of the same base.
3. Equate the exponents and solve the resulting linear (sometimes quadratic) equation.
4. If a rewritten path requires an even root of a negative, discard that path in the reals.

Same-base comparison also settles “which is larger” stems without a calculator. $4^{30} = (2^2)^{30} = 2^{60}$ and $8^{20} = (2^3)^{20} = 2^{60}$, so the two values are equal — a typical no-calculator identity check.

Factoring Exponential Sums and Differences

You cannot add bases or add exponents across a $+$ or $-$:

$2^x + 2^x \ne 4^x, \qquad 3^5 + 3^5 + 3^5 \ne 3^{15}$

Factor the lowest common power instead.

• $3^5 + 3^5 + 3^5 = 3 \cdot 3^5 = 3^6$.
• $2^x + 2^x = 2 \cdot 2^x = 2^{x+1}$.
• $3^{x+2} - 3^x = 3^x(3^2 - 1) = 8 \cdot 3^x$.
• $\dfrac{2^{20} - 2^{17}}{2^{18} + 2^{17}} = \dfrac{2^{17}(2^3 - 1)}{2^{17}(2 + 1)} = \dfrac{7}{3}$.

Worked GMAT-Style Examples

Worked Example 1: Base reduction

If $\left(\dfrac{1}{9}\right)^{2x-1} = 27^{x+4}$, find $x$.

1. $\dfrac{1}{9} = (3^2)^{-1} = 3^{-2}$ and $27 = 3^3$.
2. $(3^{-2})^{2x-1} = (3^3)^{x+4}$.
3. $3^{-4x+2} = 3^{3x+12}$.
4. $-4x + 2 = 3x + 12 \implies -7x = 10 \implies x = -\dfrac{10}{7}$.

Worked Example 2: Factor a common power

If $5^{k+2} - 5^k = 600$ and $k$ is an integer, find $k$.

1. $5^k(5^2 - 1) = 600 \implies 24 \cdot 5^k = 600$.
2. $5^k = 25 = 5^2$, so $k = 2$.

Live Quant items have five answer choices; the checks in this guide use four. The algebra does not change.