Does the GRE test logarithms?

No. Logarithms are off-syllabus on the GRE. To solve an exponential equation, rewrite both sides with the same base and set the exponents equal — for example, 2^(x+1) = 8^(x-1) becomes x + 1 = 3(x - 1).

How do you evaluate a fractional exponent like 8^(2/3)?

The denominator is the root and the numerator is the power. Take the cube root of 8 first (which is 2), then square it to get 4. Taking the root before the power keeps the numbers small and avoids arithmetic mistakes.

What does a negative exponent mean?

A negative exponent means take the reciprocal, not make the number negative. So 2^(-3) equals 1 divided by 2^3, which is 1/8. The result of a negative exponent on a positive base is always positive.

How do you compare which power is larger without a calculator?

Rewrite the powers with a common base, then compare the exponents. For example, 4^15 and 8^10 both equal 2^30, while 16^8 equals 2^32, so 16^8 is the largest. If bases cannot be matched, match the exponents instead.

CramClub • Exponents and Roots — GRE® General Test Guide & Practice

$a^m \cdot a^n = a^{m+n}, \quad \dfrac{a^m}{a^n} = a^{m-n}$

$a^{-n} = \dfrac{1}{a^n}, \quad a^0 = 1$

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

$\sqrt{ab^2} = b\sqrt{a}, \quad \dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}$

Why Exponents Dominate GRE Arithmetic

Exponents and roots are the connective tissue of GRE Quant. They surface in pure-arithmetic problems, in algebraic simplification, inside geometry (areas and volumes), and — most dangerously — in quantitative comparison, where a single mis-applied rule flips your answer. The good news is that the entire topic reduces to about seven rules plus the discipline to apply them exactly. There are no logarithms on the GRE, so you never solve for an exponent with a log; you match bases instead.

The Core Exponent Rules

An exponent is repeated multiplication: $a^n$ means $a$ multiplied by itself $n$ times. Every rule below follows from that definition.

Rule	Statement	Example
Product	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^4 = 2^7$
Quotient	$\dfrac{a^m}{a^n} = a^{m-n}$	$\dfrac{5^6}{5^2} = 5^4$
Power of a power	$(a^m)^n = a^{mn}$	$(3^2)^4 = 3^8$
Power of a product	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Zero	$a^0 = 1$ (for $a \ne 0$ )	$17^0 = 1$
Negative	$a^{-n} = \dfrac{1}{a^n}$	$2^{-3} = \dfrac{1}{8}$

The two rules that trap the most test-takers: a negative exponent means reciprocal, not a negative number ( $2^{-3} = \tfrac{1}{8}$ , never $-8$ ), and the product/quotient rules only apply when the bases are the same. You cannot combine $2^3 \cdot 3^2$ by adding exponents.

Fractional Exponents Are Roots

A fractional exponent packages a root and a power together:

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

So $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ . Take the root first and the numbers stay small. Reading a fractional exponent this way — denominator is the root, numerator is the power — is one of the highest-yield GRE skills.

Square Roots, Cube Roots, and Simplifying Radicals

To simplify a radical, factor out the largest perfect square (or cube). For example, $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$ . Radicals combine only when the part under the root matches: $5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$ , but $\sqrt{2} + \sqrt{3}$ cannot be merged. A crucial non-rule: $\sqrt{a^2 + b^2} \ne a + b$ . The root of a sum is not the sum of the roots.

Rationalizing the Denominator

The GRE prefers answers with no radical in the denominator. Multiply the fraction by the radical over itself:

$\frac{6}{\sqrt{3}} = \frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$

Scientific Notation

Very large or very small numbers are written as $c \times 10^{k}$ with $1 \le c < 10$ . Multiply the leading numbers and add the powers of ten; divide the leading numbers and subtract the powers:

$(3 \times 10^{8})(2 \times 10^{-3}) = 6 \times 10^{5}, \qquad \frac{6 \times 10^{5}}{4 \times 10^{2}} = 1.5 \times 10^{3}$

Comparing Exponential Magnitudes

When a question asks which of several powers is largest, rewrite everything with a common base (usually the smallest base). To compare $4^{15}$ , $8^{10}$ , and $2^{28}$ , convert to base 2: $4^{15} = 2^{30}$ , $8^{10} = 2^{30}$ , and $2^{28}$ — now the exponents alone decide it. If bases cannot be matched, match the exponents instead: to compare $2^{30}$ and $3^{20}$ , write $2^{30} = (2^3)^{10} = 8^{10}$ and $3^{20} = (3^2)^{10} = 9^{10}$ , so $3^{20}$ is larger.

A Reliable Game Plan

Same base, multiplied or divided? Add or subtract exponents.
A power raised to a power? Multiply the exponents.
Fractional exponent? Denominator is the root — take it first.
Radical to simplify? Pull out the largest perfect square or cube.
Radical in a denominator? Multiply top and bottom by that radical.
Comparing sizes? Force a common base, then compare exponents.

Simplify $\dfrac{x^5 \cdot x^3}{x^2}$ .

Evaluate $5^0 + 2^{-3}$ .

Simplify $\left(2x^2 y^{-1}\right)^3$ and write it with no negative exponents.

Evaluate $32^{2/5}$ .

Simplify $\sqrt{72}$ , then rationalize $\dfrac{6}{\sqrt{3}}$ .

Solve $2^{x+1} = 8^{x-1}$ for $x$ .

Treating a negative exponent as a negative number, e.g. writing $3^{-2} = -9$ .

A negative exponent means reciprocal: $3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$ . The value stays positive.

Adding exponents when the bases differ, e.g. $2^3 \cdot 3^2 = 6^5$ .

The product rule requires identical bases. $2^3 \cdot 3^2 = 8 \cdot 9 = 72$ — just evaluate each power.

Splitting the root of a sum, writing $\sqrt{a^2 + b^2} = a + b$ .

Roots do not distribute over addition. $\sqrt{9 + 16} = \sqrt{25} = 5$ , not $3 + 4 = 7$ . Combine inside the radical first.

Misreading a fractional exponent, computing $8^{2/3}$ as $8 \cdot \tfrac{2}{3}$ or as $\sqrt[2]{8^3}$ .

The denominator is the root: $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ . Root by the bottom, power by the top.

Which of the following is equivalent to $\dfrac{\left(x^3 y^{-2}\right)^2}{x^{-1} y^{3}}$ for nonzero $x$ and $y$ ?

Which of the following is the greatest?

What is the value of $\dfrac{(3 \times 10^{8})(2 \times 10^{-3})}{4 \times 10^{2}}$ ?

What is the value of $16^{3/4}$ ?

Which of the following is equal to $\sqrt{50} + \sqrt{18}$ ?

Quantity A: $27^{1/3} \cdot 4^{1/2}$

Quantity B: $6$

$a^m \cdot a^n = a^{m+n}, \quad \dfrac{a^m}{a^n} = a^{m-n}$

$a^{-n} = \dfrac{1}{a^n}, \quad a^0 = 1$

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

$\sqrt{ab^2} = b\sqrt{a}, \quad \dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}$

Why Exponents Dominate GRE Arithmetic

The Core Exponent Rules

An exponent is repeated multiplication: $a^n$ means $a$ multiplied by itself $n$ times. Every rule below follows from that definition.

Rule	Statement	Example
Product	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^4 = 2^7$
Quotient	$\dfrac{a^m}{a^n} = a^{m-n}$	$\dfrac{5^6}{5^2} = 5^4$
Power of a power	$(a^m)^n = a^{mn}$	$(3^2)^4 = 3^8$
Power of a product	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Zero	$a^0 = 1$ (for $a \ne 0$ )	$17^0 = 1$
Negative	$a^{-n} = \dfrac{1}{a^n}$	$2^{-3} = \dfrac{1}{8}$

Fractional Exponents Are Roots

A fractional exponent packages a root and a power together:

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

Square Roots, Cube Roots, and Simplifying Radicals

Rationalizing the Denominator

The GRE prefers answers with no radical in the denominator. Multiply the fraction by the radical over itself:

$\frac{6}{\sqrt{3}} = \frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$

Scientific Notation

Very large or very small numbers are written as $c \times 10^{k}$ with $1 \le c < 10$ . Multiply the leading numbers and add the powers of ten; divide the leading numbers and subtract the powers:

$(3 \times 10^{8})(2 \times 10^{-3}) = 6 \times 10^{5}, \qquad \frac{6 \times 10^{5}}{4 \times 10^{2}} = 1.5 \times 10^{3}$

Comparing Exponential Magnitudes

A Reliable Game Plan

Same base, multiplied or divided? Add or subtract exponents.
A power raised to a power? Multiply the exponents.
Fractional exponent? Denominator is the root — take it first.
Radical to simplify? Pull out the largest perfect square or cube.
Radical in a denominator? Multiply top and bottom by that radical.
Comparing sizes? Force a common base, then compare exponents.