Quant · Basic concepts · 02
Basic math concept 02: exponents and the exponent laws
Exponents are shorthand for repeated multiplication, and a small set of laws governs how they combine. Learn the laws once and most exponent questions on the GRE and GMAT become a matter of matching a rule. This is concept 02 in the basic-math series.
These are standard arithmetic rules; the skills they support are checked against the GRE Quantitative Reasoning overview from ETS. The byline is a masthead name, explained in full on who writes this, and the sourcing rules are set out on the standards page.
The whole concept rests on one definition: a^n means a multiplied by itself n times, so 2^3 = 2 × 2 × 2 = 8. Everything below follows from unpacking that single idea. If you ever forget a law, write the exponents out as repeated multiplication and the rule will reappear.
The six laws
| Law | Rule | What it means |
|---|---|---|
| Product of powers | a^m × a^n = a^(m+n) | Same base, add the exponents. 2^3 × 2^4 = 2^7. |
| Quotient of powers | a^m ÷ a^n = a^(m−n) | Same base, subtract the exponents. 2^5 ÷ 2^2 = 2^3. |
| Power of a power | (a^m)^n = a^(m×n) | Multiply the exponents. (2^3)^2 = 2^6 = 64. |
| Power of a product | (ab)^n = a^n × b^n | The exponent applies to each factor. (2x)^3 = 8x^3. |
| Zero exponent | a^0 = 1 (for a ≠ 0) | Any nonzero base to the zero power is 1. |
| Negative exponent | a^(−n) = 1 / a^n | A negative exponent means reciprocal, not a negative number. 2^(−3) = 1/8. |
Worked examples
- Simplify x^3 × x^4. Same base, add exponents: x^(3+4) = x^7.
- Simplify (x^2)^5. Power of a power, multiply: x^(2×5) = x^10.
- Simplify 2^−3. Negative exponent means reciprocal: 1 / 2^3 = 1/8.
- Simplify (3x)^2. Power of a product applies to each factor: 3^2 × x^2 = 9x^2.
- Simplify y^7 ÷ y^7. Same base, subtract: y^0, which equals 1.
The three traps the tests set
- Negative exponents are not negative numbers. 2^(−3) is +1/8, not −8. The exponent is negative; the value is a small positive fraction. Test writers bury this inside larger expressions to lure you toward a sign error.
- Adding exponents only works when you multiply, not when you add. x^2 + x^3 does not simplify to x^5. You can only combine exponents across multiplication or division of the same base, never across addition.
- Negative bases need parentheses. (−2)^4 = 16, but −2^4 = −16. Without parentheses, the exponent binds before the sign. The GRE exploits this distinction relentlessly.
A fractional exponent is a root in disguise. a^(1/2) is the square root of a, a^(1/3) is the cube root, and a^(2/3) is the cube root of a squared. You do not need to go deep here for the basics, but knowing the translation stops a fractional exponent from looking alien later.
Pair this concept with the quant formula sheet, which lists the exponent rules alongside everything else, and test yourself in thepractice bank, where questions likeadding exponents andpower cycles put these laws to work.