Is ##9\sqrt[3]{-3}## Equivalent to ##-9\sqrt[3]{3}##?

[RChristenk]

Homework Statement

Find $\sqrt[3]{-2187}$

Relevant Equations

None

I calculated this to be $9\sqrt[3]{-3}$, but the answer is given as $-9\sqrt[3]{3}$. Are these two quantities equal? If so, what is the usual convention for placement of the negative sign? Thanks.

 

[.Scott]

When you take a cube root or other odd root of -1, you get -1 (at least until you get to the chapter on complex numbers). So, in those cases, moving the sign outside the radical is considered a simplification.

 

[Drakkith]

Just FYI, I changed the title from square root to cube root.

 

[PeroK]

Homework Statement: Find $\sqrt[3]{-2187}$
Relevant Equations: None

I calculated this to be $9\sqrt[3]{-3}$, but the answer is given as $-9\sqrt[3]{3}$. Are these two quantities equal? If so, what is the usual convention for placement of the negative sign? Thanks.

You could look at a graph of the cube root function:

https://www.cuemath.com/calculus/cube-root-function/

And you'll see that the cube root of a negative number is just a regular negative number.

 

[Mark44]

A simpler example is this: $\sqrt[3]{-8} = -2$. As a check, cube the result on the right side.
$(-2)^3 = (-2)(-2)(-2) = -8$