Exponents and Roots

Concise answer

Every exponent rule assumes a shared base or a shared exponent: add exponents when multiplying powers of the same base, multiply exponents when raising a power to a power, read negative exponents as reciprocals and fractional exponents as roots — and never distribute an exponent over a sum.

Definitions

Exponent: The number of times a base is used as a factor; in $a^n$, $a$ is the base and $n$ is the exponent.
Negative exponent: For nonzero $a$, $a^{-n}$ means the reciprocal $\tfrac{1}{a^n}$.
Rational (fractional) exponent: For suitable $a$, $a^{1/n}$ means the $n$th root of $a$, and $a^{m/n}$ means that root raised to the $m$th power.
Rationalizing the denominator: Rewriting a fraction so no radical remains in the denominator, by multiplying the numerator and denominator by an appropriate radical.

Intuition

An exponent counts repeated multiplication, so multiplying $a^m$ by $a^n$ just strings the factors together — which is why the exponents add rather than multiply.

Negative and fractional exponents extend that counting pattern: stepping the exponent down by 1 divides by the base once (so $a^{-n}$ lands at a reciprocal), and $a^{1/2}$ must be the number whose square is $a$ .

Concept walkthrough

The core exponent rules all require a shared base: $a^m \cdot a^n = a^{m+n}$ , $\tfrac{a^m}{a^n} = a^{m-n}$ , and $(a^m)^n = a^{mn}$ . Combined with $(ab)^n = a^n b^n$ for a shared exponent, these four identities cover almost every GRE exponent manipulation. When neither the bases nor the exponents match, no rule applies — the first move on many problems is rewriting terms so a common base appears, such as $4^x = (2^2)^x = 2^{2x}$ .

Negative and fractional exponents are notational extensions of the same system. For nonzero $a$ , $a^0 = 1$ and $a^{-n} = \tfrac{1}{a^n}$ , so a negative exponent signals a reciprocal, not a negative number. A fractional exponent signals a root: $a^{1/2} = \sqrt{a}$ and $a^{m/n} = (\sqrt[n]{a})^m$ . On the GRE these appear together, so an expression like $8^{2/3}$ is best read in two steps: cube root of $8$ is $2$ , then $2^2 = 4$ .

Roots follow mirror-image rules: $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ for nonnegative $a$ and $b$ , which is how $\sqrt{50}$ simplifies to $5\sqrt{2}$ . Answer choices are conventionally written without radicals in the denominator, so a value like $\tfrac{6}{\sqrt{3}}$ is rationalized by multiplying by $\tfrac{\sqrt{3}}{\sqrt{3}}$ to get $\tfrac{6\sqrt{3}}{3} = 2\sqrt{3}$ . ETS lists exponents and roots in the arithmetic content of the Quantitative Reasoning measure, so these manipulations are tested directly and inside larger algebra problems.

After this page, you should be able to

Apply the product, quotient, and power properties of exponents to expressions with a common base.
Rewrite negative exponents as reciprocals and fractional exponents as roots.
Simplify square roots and rationalize a denominator containing a radical.
Recognize which combinations of bases and exponents the rules do not cover.

Formulas and assumptions

Product and power properties of exponents

a^m \cdot a^n = a^{m+n} \qquad (a^m)^n = a^{mn}

Variables

a: the common base
m, n: the exponents

Assumptions

Both powers share the same base a.
If an exponent is negative or zero, a is not zero.

Negative and fractional exponents

a^{-n} = \dfrac{1}{a^n} \qquad a^{m/n} = \left(\sqrt[n]{a}\right)^{m}

Variables

a: the base
n: the (positive) root index or exponent magnitude
m: the power applied after taking the root

Assumptions

a is not zero when the exponent is negative or zero.
For even roots, a is nonnegative so the expression stays a real number.

Worked example

Collapsing an expression to one power of 2

Simplify $\left(2^3 \cdot 2^{-5}\right)^{-2}$ .

1Inside the parentheses the bases match, so add the exponents: $2^3 \cdot 2^{-5} = 2^{3 + (-5)} = 2^{-2}$ .
2Raise the power to a power by multiplying the exponents: $\left(2^{-2}\right)^{-2} = 2^{(-2)(-2)} = 2^{4}$ .
3Evaluate: $2^4 = 16$ .
4Sanity-check with the reciprocal reading: $2^{-2} = \tfrac{1}{4}$ , and $\left(\tfrac{1}{4}\right)^{-2} = 4^2 = 16$ , which agrees.

$\left(2^3 \cdot 2^{-5}\right)^{-2} = 16$ .

Common traps

Distributing an exponent over a sum: $(a + b)^2$ is $a^2 + 2ab + b^2$ , never $a^2 + b^2$ , and $\sqrt{a^2 + b^2}$ is not $a + b$ .
Reading a negative exponent as a negative value: $2^{-3} = \tfrac{1}{8}$ , a positive number.
Multiplying exponents when multiplying powers ( $2^3 \cdot 2^4$ is $2^7$ , not $2^{12}$ ) or adding them when raising a power to a power.
Dropping parentheses with negative bases: $(-3)^2 = 9$ but $-3^2 = -9$ , and for a variable, $\sqrt{x^2} = |x|$ , not $x$ .

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Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

GRE General Test Quantitative Reasoning Overview — ETS. Accessed 2026-08-02. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
Elementary Algebra 2e, Section 6.2: Use Multiplication Properties of Exponents — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Elementary Algebra 2e, Section 6.7: Integer Exponents and Scientific Notation — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Elementary Algebra 2e, Section 9.5: Divide Square Roots — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Intermediate Algebra 2e, Section 8.3: Simplify Rational Exponents — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.

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Last source check: 2026-08-02
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