Algebra › Exponents & roots
Exponents & roots
A small number perched up high tells you how many times to multiply. A square root undoes squaring. The whole topic is repeated multiplication and its reverse.
Build an exponent
An exponent says: multiply the base by itself this many times.
baseexp
24=16
Try a negative exponent — it flips into 1 over the positive version. Try 0 — anything to the zeroth power is 1.
What an exponent means
2³ means "multiply 2 by itself three times": 2 × 2 × 2 = 8. The big number on the bottom is the base; the little number up high is the exponent.
The five laws you keep using
- Same base, multiply — add the exponents.
aᵐ × aⁿ = aᵐ⁺ⁿ - Same base, divide — subtract the exponents.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ - Power of a power — multiply the exponents.
(aᵐ)ⁿ = aᵐⁿ - Power of a product — distribute the exponent.
(ab)ⁿ = aⁿ bⁿ - Zero exponent — anything (except 0) to the 0 is 1.
a⁰ = 1
Negative exponents flip
a⁻ⁿ = 1 / aⁿ. So 2⁻³ = 1/8. A negative exponent is the reciprocal of the positive version.Roots — the inverse
The square root undoes squaring: √25 = 5 because 5² = 25. The cube root undoes cubing: ∛27 = 3. In general, ⁿ√a is the number you must raise to the n-th power to get a back.
Fractional exponents
A fractional exponent is just a root in disguise. a^(1/2) = √a, a^(1/3) = ∛a, and a^(m/n) = ⁿ√(aᵐ). Once you see this, every root is just another exponent rule.
Quick check
- Simplify
x³ × x⁵. - Simplify
(2y²)³. - Write
1/16as a power of 2.
Answers: x⁸, 8y⁶, and 2⁻⁴.
Simplify x³ × x⁵.
Write 1/16 as a power of 2.