Meaning of Rational Exponents
Aligned to N-RN.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
Rational exponents are defined so the rule a^m × a^n = a^(m+n) keeps working. If a^(1/2) × a^(1/2) must equal a^1 = a, then a^(1/2) has to be √a, and in general a^(1/n) is the nth root of a.
Worked examples
Example 1
Why does 8^(1/3) = 2? By the exponent rule, (8^(1/3))^3 = 8^(3/3) = 8^1 = 8. So 8^(1/3) is the number that cubes to 8, which is 2. This forces a^(1/3) to mean the cube root.
Example 2
Why does 9^(1/2) = 3? The exponent rule (a^m)^n = a^(mn) must keep working, so (9^(1/2))² = 9^(2/2) = 9. That means 9^(1/2) is a number whose square is 9, which is 3. The same argument makes a^(1/n) the nth root: it is the only definition that preserves the rule.
Example 3
Common mistake: expecting a^(1/n) to be defined for every base. When n is even, a negative base has no real value: (−16)^(1/2) would need a real number squaring to −16, and none exists. When n is odd it works fine: (−8)^(1/3) = −2, because (−2)³ = −8. Check whether the root index is even before assuming a real answer exists.
Practice questions
- Explain why 25^(1/2) must equal 5, not 12.5.
- Use the exponent rules to show (a^(1/4))^4 = a.
- What number must 27^(1/3) equal, and why?
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
Significado de los Exponentes Racionales
Los exponentes racionales se definen para que la regla a^m × a^n = a^(m+n) siga funcionando. Si a^(1/2) × a^(1/2) debe ser igual a a^1 = a, entonces a^(1/2) tiene que ser √a, y en general a^(1/n) es la raíz enésima de a.
Ejemplo 1: ¿Por qué 8^(1/3) = 2? Por la regla de exponentes, (8^(1/3))^3 = 8^(3/3) = 8^1 = 8. Así que 8^(1/3) es el número que elevado al cubo da 8, es decir, 2. Esto obliga a que a^(1/3) signifique la raíz cúbica.
Ejemplo 2: ¿Por qué 9^(1/2) = 3? La regla de exponentes (a^m)^n = a^(mn) debe seguir funcionando, así que (9^(1/2))² = 9^(2/2) = 9. Eso significa que 9^(1/2) es un número cuyo cuadrado es 9, es decir 3. El mismo argumento hace que a^(1/n) sea la raíz n-ésima: es la única definición que conserva la regla.
Ejemplo 3: Error común: esperar que a^(1/n) esté definido para cualquier base. Cuando n es par, una base negativa no tiene valor real: (−16)^(1/2) necesitaría un número real cuyo cuadrado fuera −16, y no existe. Cuando n es impar sí funciona: (−8)^(1/3) = −2, porque (−2)³ = −8. Revisa si el índice de la raíz es par antes de suponer que hay respuesta real.
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