Algebra

3*cbrt(-81x^9)

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Identify the cube root expression

We need to simplify 3 times the cube root of negative 81x⁹. We'll break the expression inside the cube root into perfect cube factors.
2

Factor -81 into cube and leftover

-27 is a perfect cube because (-3)³ = -27. We write -81 as -27 times 3 so we can pull -27 out of the cube root.
3

Factor x⁹ as a perfect cube

Since 9 divided by 3 equals 3, x⁹ is a perfect cube. The cube root of x⁹ is simply x³.
4

Rewrite inside the cube root

We split the contents of the cube root into the perfect cube parts (-27 and (x³)³) and the leftover factor (3).
5

Pull perfect cubes outside the root

The cube root of -27 is -3, and the cube root of (x³)³ is x³. These come out of the radical. The leftover factor of 3 stays inside.
6

Multiply the coefficients together

Multiply 3 by -3 to get -9. The final simplified expression is -9x³ times the cube root of 3.

Understanding this problem

Learning Insight

Cube roots and cube powers 'undo' each other. When you factor out perfect cubes from inside a radical, you're using the rule ∛(a³) = a. Breaking numbers into perfect-cube factors is the key strategy for simplifying any cube root.

Quick Tip

For any number, quickly ask: what perfect cube (8, 27, 64, 125, …) divides it evenly? That factor comes outside the cube root. For variables, divide the exponent by 3 — the quotient goes outside and the remainder stays inside.

Common Mistake

Students often try to take the cube root of -81 directly without factoring first, and get stuck because -81 isn't a perfect cube. Always factor out the largest perfect cube first (here, -27), then handle the leftover.