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**Hint:**For solving this question you should know about the cube root of a number. In this problem we will solve the cube root of this number by long division method because it is more accurate and not so much of a lengthy method and it provides us the exact answer whether our number is fully a cube or not.

**Complete step-by-step solution:**

According to our question, we have been asked to find the cube root of -27000. As we know for solving using the long division method, we use just simple steps as follows:

Step 1: Divide our given term by which it is asked and it will be a simple division.

Step 2: Now we will multiply the quotient with the divider and put the value of that at subtraction of the first digit.

Step 3: Then we put the remaining single digit with the result of this subtraction and then we get it as a new number.

Step 4: Now we get a new number. Apply the same above steps with it and finally get the last remainder which can’t be divisible by anything else or we get a remainder as 0.

Step 5: If you want to check this, then you can do so with the formula:

Dividend = Divisor $\times $ Quotient + Remainder

Now if we look at our question, $\sqrt[3]{-27000}$ by long division method:

$\begin{align}

& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,30 \\

& \,\,\,\,\,\,\,\,\,9\left| \!{\overline {\,

\begin{align}

& 27000 \\

& 27 \\

\end{align} \,}} \right. \\

& 2700\left| \!{\overline {\,

\begin{align}

& \,\,\,\,\,\,\,\,\,\,\,\,0 \\

& \,\,\,\,\,\,\,\,\,\,\,\,0 \\

\end{align} \,}} \right. \\

& \,\,\,\,...\left| \!{\overline {\,

\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \,}} \right. \\

\end{align}$

$300\times {{3}^{2}}+30\times 3\times 0+{{0}^{2}}=2700$ And as we know if we make a cube of -30, then we get,

$\begin{align}

& =\left( -30 \right)\left( -30 \right)\left( -30 \right)=\left( -30 \right)\left( 900 \right) \\

& =-27000 \\

\end{align}$

**So, the cube root is -30.**

**Note:**While solving this question or all other related questions to the long division method, always try to make it fully divided or make the remainder zero. And if the question is asked for the square root, then it is mandatory for the remainder to be zero. If this will be zero, then maybe your calculations are wrong and you will never get the solution right.

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