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

See a solution process below:

This expression can be rewritten as:

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

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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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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.

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.

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.

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