Exponents & Roots
GRE Quantitative
Master the laws of exponents, rational powers, and radical expressions that appear throughout the GRE Quantitative section.
GRE Quantitative
What Are Exponents?
An exponent tells you how many times to multiply a number (the base) by itself. In the expression \( a^n \), the base is \( a \) and the exponent is \( n \).
What Are Roots?
A root is the inverse of an exponent. The \( n \)-th root of \( a \), written \( \sqrt[n]{a} \), asks: "What number raised to the \( n \)-th power gives \( a \)?"
Why Exponents & Roots Matter on the GRE
Quantitative Comparison
Comparing expressions like \( 2^{10} \) vs. \( 4^5 \) requires rewriting bases and applying exponent rules.
Simplification
Many GRE problems hinge on simplifying nested radicals or combining exponents to reach a clean answer.
Data Interpretation
Scientific notation (\( 3.2 \times 10^6 \)) and percentage growth rely on exponent fluency.
Core Laws of Exponents
1
Product Rule: \( a^m \cdot a^n = a^{m+n} \)
2
Quotient Rule: \( \dfrac{a^m}{a^n} = a^{m-n} \)
3
Power Rule: \( (a^m)^n = a^{mn} \)
4
Product-to-Power: \( (ab)^n = a^n \cdot b^n \)
5
Quotient-to-Power: \( \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \)
Special Exponent Values
1
Zero exponent: \( a^0 = 1 \) for any \( a \neq 0 \)
2
Negative exponent: \( a^{-n} = \dfrac{1}{a^n} \)
3
Fractional exponent: \( a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \)
Core Root Laws
1
Product of Roots: \( \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} \)
2
Quotient of Roots: \( \sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} \)
3
Root of a Root: \( \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} = a^{1/(mn)} \)
4
Rationalizing: Multiply by \( \dfrac{\sqrt{a}}{\sqrt{a}} \) to remove radicals from denominators.
Exponents vs. Roots: Key Distinction
| Exponents | Roots | |
|---|---|---|
| Notation | \( a^n \) | \( \sqrt[n]{a} \) or \( a^{1/n} \) |
| Operation | Repeated multiplication | Inverse of exponentiation |
| Example | \( 3^4 = 81 \) | \( \sqrt[4]{81} = 3 \) |
| Combine | Add exponents (same base) | Convert to fractional exponents, then add |
| GRE Focus | Simplify & compare quantities | Simplify radicals & rationalize |
How to Simplify Exponent Expressions
1
Rewrite all bases as prime factors (e.g., \( 8 = 2^3 \), \( 27 = 3^3 \)).
2
Convert roots to fractional exponents: \( \sqrt[n]{a} \to a^{1/n} \).
3
Apply exponent laws: combine like bases (add, subtract, or multiply exponents).
4
Simplify the result. Convert back to radical form if the answer choices use radicals.
5
Check sign & domain. Negative bases with even exponents are positive. Even roots require non-negative radicands.
Example 1: Product Rule
Simplify \( 2^3 \cdot 2^5 \).
Example 2: Fractional Exponents
Simplify \( 27^{2/3} \).
Example 3: Negative Exponents & Quotients
Simplify \( \dfrac{4^{-2} \cdot 4^5}{4^2} \).
1
Numerator: Combine using the product rule: \( 4^{-2} \cdot 4^5 = 4^{-2+5} = 4^3 \).
2
Quotient rule: \( \dfrac{4^3}{4^2} = 4^{3-2} = 4^1 = 4 \).
Example 4: Simplifying Radicals
Simplify \( \sqrt{50} + \sqrt{18} \).
1
Factor under the radical: \( \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \).
2
Factor the second radical: \( \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \).
3
Combine like radicals: \( 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2} \).
Visualising \( y = x^n \)
Common Mistakes to Avoid
✕ Adding Exponents Wrong
\( a^m + a^n \neq a^{m+n} \). The product rule only applies to multiplication, not addition.
✕ Distributing Exponents Over Sums
\( (a + b)^2 \neq a^2 + b^2 \). You must FOIL: \( (a+b)^2 = a^2 + 2ab + b^2 \).
✕ Negative Base Confusion
\( -3^2 = -9 \) but \( (-3)^2 = 9 \). Parentheses around the base matter.
Summary of Exponent & Root Rules
| Rule | Expression | Result |
|---|---|---|
| Product | \( a^m \cdot a^n \) | \( a^{m+n} \) |
| Quotient | \( a^m / a^n \) | \( a^{m-n} \) |
| Power | \( (a^m)^n \) | \( a^{mn} \) |
| Zero Exp. | \( a^0 \) | \( 1 \) |
| Negative Exp. | \( a^{-n} \) | \( 1/a^n \) |
| Fractional Exp. | \( a^{m/n} \) | \( \sqrt[n]{a^m} \) |
| Root Product | \( \sqrt{a} \cdot \sqrt{b} \) | \( \sqrt{ab} \) |
Question 1 of 5
Simplify: \( 3^4 \cdot 3^{-2} \)
Using the product rule: \( 3^4 \cdot 3^{-2} = 3^{4+(-2)} = 3^2 = 9 \). Add the exponents when the bases are the same.
Question 2 of 5
What is \( 8^{2/3} \)?
Rewrite: \( 8^{2/3} = (2^3)^{2/3} = 2^{3 \cdot 2/3} = 2^2 = 4 \). Express 8 as \( 2^3 \), then multiply exponents.
Question 3 of 5
Simplify: \( \sqrt{72} \)
\( \sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \). Factor out the largest perfect square (36) from under the radical.
Question 4 of 5
Which expression equals \( \dfrac{1}{16} \)?
\( 2^{-4} = \dfrac{1}{2^4} = \dfrac{1}{16} \). Note: \( 4^{-1} = \frac{1}{4} \), \( 2^{-3} = \frac{1}{8} \), and \( 16^{-2} = \frac{1}{256} \).
Question 5 of 5
If \( 5^x = 125 \), what is \( 5^{x-1} \)?
Since \( 5^x = 125 = 5^3 \), we know \( x = 3 \). Then \( 5^{x-1} = 5^{3-1} = 5^2 = 25 \). Alternatively, \( 5^{x-1} = \frac{5^x}{5} = \frac{125}{5} = 25 \).