Rational Exponents
Aligned to N-RN.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
To keep the rule (a^m)^n = a^(m×n) working for fractions, we define a^(1/n) as the number whose nth power is a. So a^(1/n) is the nth root of a, and a^(m/n) means (nth root of a)^m.
Worked examples
Example 1
Evaluate 8^(2/3). First 8^(1/3) = 2, because 2^3 = 8. Then 8^(2/3) = (8^(1/3))^2 = 2^2 = 4.
Example 2
Evaluate 16^(3/4). Take the fourth root first: 16^(1/4) = 2, then 2³ = 8.
Example 3
A negative exponent is a reciprocal, not a negative number: 8^(−2/3) = 1/4.
Practice questions
- Write 5^(1/2) as a radical.
- Evaluate 27^(1/3).
- Evaluate 16^(3/4).
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Exponentes Racionales
Para que la regla (a^m)^n = a^(m×n) siga funcionando con fracciones, definimos a^(1/n) como el número cuya potencia n es a. Así, a^(1/n) es la raíz n de a, y a^(m/n) significa (raíz n de a)^m.
Ejemplo 1: Evalúa 8^(2/3). Primero 8^(1/3) = 2, porque 2^3 = 8. Luego 8^(2/3) = (8^(1/3))^2 = 2^2 = 4.
Ejemplo 2: Evalúa 16^(3/4). Primero la raíz cuarta: 16^(1/4) = 2, luego 2³ = 8.
Ejemplo 3: Un exponente negativo es un recíproco, no un número negativo: 8^(−2/3) = 1/4.
More Number & Quantity lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required