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 example
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.
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?
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Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
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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: ¿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.
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