How do you simplify #root(3)(-1080)#?

Answer 1

Everly Taylor

See a solution process below:

This expression can be rewritten as:

#root(3)(-8 xx 27 xx 5)#

The expression can then be made simpler by applying the following radical rule:

#root(n)(color(red)(a) * color(blue)(b)) = root(n)(color(red)(a)) * root(n)(color(blue)(b))#

#root(3)(color(red)(-8) * color(blue)(27) * 5) =>#

#root(3)(color(red)(-8)) * root(3)(color(blue)(27)) * root(3)(5) =>#

#-2 * 3 * root(3)(5) =>#

#-6root(3)(5)#

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Answer 2

Jeremiah Andrews

To simplify $\sqrt[3]{-1080}$ , we first need to determine whether -1080 has a real cube root. Since -1080 is negative, it does not have a real cube root. However, it does have a complex cube root.

The cube root of -1080 can be written as:

$\sqrt[3]{-1080} = \sqrt[3]{-1} \times \sqrt[3]{1080}$

$= -1 \times \sqrt[3]{1080}$

Now, we need to simplify $\sqrt[3]{1080}$ .

The prime factorization of 1080 is $2^3 \times 3^3 \times 5$ .

Therefore,

$\sqrt[3]{1080} = \sqrt[3]{2^3 \times 3^3 \times 5}$

$= 2 \times 3 \times \sqrt[3]{5}$

$= 6 \sqrt[3]{5}$

Now, substituting this value into the expression for $\sqrt[3]{-1080}$ , we get:

$\sqrt[3]{-1080} = -1 \times 6 \sqrt[3]{5}$

$= -6 \sqrt[3]{5}$

So, the simplified form of $\sqrt[3]{-1080}$ is $-6 \sqrt[3]{5}$ .

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.