Combining exponents
Also known as: exponent rules, laws of exponents
Combining exponents means simplifying expressions with powers using the exponent rules: add exponents when multiplying like bases, subtract when dividing, and multiply when raising a power to a power.
Combining exponents applies a small set of rules to simplify expressions with powers. The three workhorses: the product rule, x^a · x^b = x^(a+b); the quotient rule, x^a / x^b = x^(a−b); and the power rule, (x^a)^b = x^(ab). So x³ · x⁴ = x⁷, x⁵ / x² = x³, and (x²)³ = x⁶.
The rules only combine powers that share the same base. You can simplify 2³ · 2⁴ to 2⁷, but 2³ · 5⁴ stays as it is — the bases differ. When different bases each carry the same exponent, a separate rule applies: (xy)^a = x^a · y^a, so (2x)³ = 8x³. A frequent trick on test questions is rewriting bases to match: 4^x · 2^x becomes 2^(2x) · 2^x = 2^(3x).
A few special cases complete the toolkit. Any nonzero base to the zero power equals 1. A negative exponent means a reciprocal: x^(−2) = 1 / x². A fractional exponent is a root: x^(1/2) is the square root of x. These let you rewrite radical and fraction-heavy expressions into a single power that the combining rules can handle.
The most common error is confusing rules — adding exponents when the power rule calls for multiplying, or combining unlike bases. The SAT tests exponent simplification directly in its algebra sections, often mixing the rules within one expression, so drill until choosing the right rule is automatic.
Key takeaways
- Multiplying like bases adds exponents: x^a · x^b = x^(a+b).
- Dividing like bases subtracts exponents: x^a / x^b = x^(a−b).
- A power raised to a power multiplies exponents: (x^a)^b = x^(ab).
- Rules apply only to identical bases — rewrite bases (like 4 = 2²) to match when possible.
- Zero exponents give 1, negative exponents give reciprocals, and fractional exponents are roots.