Radicals and Rational Exponents
Aligned to N-RN.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
Any radical can be written as a rational exponent: the nth root of a^m equals a^(m/n). Switching to exponent form lets you simplify using the ordinary rules for exponents.
Worked examples
Example 1
Simplify 16^(3/4). Write it as (16^(1/4))^3. Since 16^(1/4) = 2 (because 2^4 = 16), this becomes 2^3 = 8. So 16^(3/4) = 8.
Example 2
Simplify 27^(−2/3). The negative exponent means take the reciprocal: 1/27^(2/3). Now 27^(1/3) = 3, so 27^(2/3) = 3² = 9. The answer is 1/9. Taking the root before the power keeps the numbers small — 27² = 729 works too, but you then need the cube root of 729.
Example 3
Common mistake: adding exponents when the bases differ. 2³ × 5³ is not 10⁶ — the rule a^m × a^n = a^(m+n) needs the SAME base. With the same exponent instead you may combine the bases: 2³ × 5³ = (2 × 5)³ = 1,000. And 2³ × 2⁵ = 2⁸ = 256, not 4⁸. Check what is shared, the base or the exponent, before combining.
Practice questions
- Rewrite √(x^3) using a rational exponent.
- Evaluate 32^(2/5).
- Simplify (x^(1/2) × x^(1/3)) as a single power of x.
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.
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Radicales y Exponentes Racionales
Todo radical se puede escribir como exponente racional: la raíz enésima de a^m es igual a a^(m/n). Cambiar a la forma de exponente permite simplificar usando las reglas comunes de exponentes.
Ejemplo 1: Simplifica 16^(3/4). Escríbelo como (16^(1/4))^3. Como 16^(1/4) = 2 (porque 2^4 = 16), queda 2^3 = 8. Así que 16^(3/4) = 8.
Ejemplo 2: Simplifica 27^(−2/3). El exponente negativo indica tomar el recíproco: 1/27^(2/3). Ahora 27^(1/3) = 3, así que 27^(2/3) = 3² = 9. La respuesta es 1/9. Tomar la raíz antes que la potencia mantiene los números pequeños — 27² = 729 también sirve, pero luego hace falta la raíz cúbica de 729.
Ejemplo 3: Error común: sumar exponentes cuando las bases son distintas. 2³ × 5³ no es 10⁶ — la regla a^m × a^n = a^(m+n) exige la MISMA base. Con el mismo exponente, en cambio, sí puedes combinar las bases: 2³ × 5³ = (2 × 5)³ = 1,000. Y 2³ × 2⁵ = 2⁸ = 256, no 4⁸. Revisa qué se comparte, la base o el exponente, antes de combinar.
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