Radicals and Rational Exponents
Aligned to N-RN.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
A radical and a rational exponent name the same thing: √a = a^(1/2) and the nth root of a^m = a^(m/n). Once written as exponents, use the exponent rules to combine or simplify.
Worked examples
Example 1
Simplify a^(1/2) × a^(1/3). Multiplying like bases adds exponents: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, so the product is a^(5/6).
Example 2
Simplify (a^(2/3))^(3/4). Powers of powers multiply: (2/3)(3/4) = 1/2, so the result is a^(1/2) = √a.
Example 3
√a + √b is not √(a+b). Test it: √9 + √16 = 7, but √25 = 5.
Practice questions
- Rewrite √(x^3) using a rational exponent.
- Simplify x^(3/4) × x^(1/4).
- Rewrite 7^(2/5) as a radical.
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
Radicales y Exponentes Racionales
Un radical y un exponente racional nombran lo mismo: √a = a^(1/2) y la raíz n de a^m = a^(m/n). Una vez escritos como exponentes, usa las reglas de los exponentes para combinar o simplificar.
Ejemplo 1: Simplifica a^(1/2) × a^(1/3). Al multiplicar bases iguales se suman los exponentes: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, así que el producto es a^(5/6).
Ejemplo 2: Simplifica (a^(2/3))^(3/4). Las potencias de potencias se multiplican: (2/3)(3/4) = 1/2, así que el resultado es a^(1/2) = √a.
Ejemplo 3: √a + √b no es √(a+b). Pruébalo: √9 + √16 = 7, pero √25 = 5.
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