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

Question

Everly Taylor · Accepted Answer

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)#

Jeremiah Andrews · Answer

undefined 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} $.