Simplifying nth Root Radicals
Grade 9 students in California Reveal Math Algebra 1 learn to simplify nth root radicals by factoring out perfect nth powers from the radicand. The rule is: the nth root of (a^n · b) equals a · (nth root of b). A radical is in simplest form when the radicand has no perfect nth power factors other than 1. Examples include cube root of 54 = cube root of (27·2) = 3·cube root of 2, fourth root of 48 = fourth root of (16·3) = 2·fourth root of 3, and cube root of -250 = cube root of (-125·2) = -5·cube root of 2.
Key Concepts
To simplify a radical expression with index $n$, rewrite the radicand by factoring out any perfect $n$th powers. If $a \geq 0$, then:.
$$\sqrt[n]{a^n \cdot b} = a \cdot \sqrt[n]{b}$$.
Common Questions
How do you simplify a radical with index n?
Factor the radicand to identify any perfect nth power factors. Separate using the Product Property of Roots: the nth root of (a^n · b) = a · (nth root of b). The result is simplified when no perfect nth power factors remain.
How do you simplify the cube root of 54?
Factor 54 as 27·2 where 27=3^3 is a perfect cube. Then cube root of 54 = cube root of (27·2) = cube root of 27 · cube root of 2 = 3·cube root of 2.
How do you simplify the fourth root of 48?
Factor 48 as 16·3 where 16=2^4 is a perfect fourth power. Then fourth root of 48 = fourth root of (16·3) = fourth root of 16 · fourth root of 3 = 2·fourth root of 3.
Can you simplify a cube root of a negative number?
Yes. Cube root of -250 = cube root of (-125·2) = -5·cube root of 2, since -125=(-5)^3 is a perfect cube.
When is a radical expression in simplest form?
A radical expression is in simplest form when the radicand contains no perfect nth power factors other than 1.
Which unit covers nth root radicals in Algebra 1?
This skill is from Unit 7: Exponents and Roots in California Reveal Math Algebra 1, Grade 9.