Question

Cube of (-3) is: -
(a) +27
(b) -9
(c) -6
(d) -27

Answer 1

Hint: First understand the meaning of cube of a number. Now, write the given number by separating the positive and negative part as \[\left( -3 \right)=\left( -1 \right)\times 3\]. Multiply -1 three times with itself and 3 three times with itself. Finally, multiply the two results together to get the answer. Use the formula: - \[{{\left( -1 \right)}^{n}}=1\], if ‘n’ is even and \[{{\left( -1 \right)}^{n}}=1\], if ‘n’ is odd.

Complete step-by-step solution
Here, we have been provided with an integer (-3) and we have to find its cube. Let us see what is a cube of a number.
In arithmetic and algebra, the cube of a number ‘n’ is its third power, that is, the result of multiplying three instances of ‘n’ together. The cube of a number or any mathematical expression is denoted by a subscript 3. For example: - \[{{5}^{3}}=125\] or \[{{\left( y+1 \right)}^{3}}\]. The cube of a number ‘n’ can also be written as the multiplication of ‘n’ with its square.
\[\Rightarrow {{n}^{3}}=n\times {{n}^{2}}\]
Now, let us come to the question. We have a negative integer given as (-3) whose cube is to be determined. So, we can write the given number by separating its positive and negative part like: -
\[\Rightarrow -3=\left( -1 \right)\times 3\]
Cubing both sides, we get,
\[\begin{align}
  & \Rightarrow {{\left( -3 \right)}^{3}}={{\left( -1 \right)}^{3}}\times {{3}^{3}} \\
 & \Rightarrow {{\left( -3 \right)}^{3}}=\left\{ \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right) \right\}\times \left\{ 3\times 3\times 3 \right\} \\
 & \Rightarrow {{\left( -3 \right)}^{3}}=\left\{ \left( -1 \right)\times \left( -1 \right)\times \left( -1 \right) \right\}\times 27 \\
\end{align}\]
Now, we know that, \[{{\left( -1 \right)}^{n}}=1\] when ‘n’ is even, \[{{\left( -1 \right)}^{n}}=-1\] when ‘n’ is odd. So, here we have \[{{\left( -1 \right)}^{3}}\] and n = 3 is an odd number. Therefore, we have \[{{\left( -1 \right)}^{3}}=-1\].
\[\Rightarrow {{\left( -3 \right)}^{3}}=\left( -1 \right)\times 27\]
\[\Rightarrow {{\left( -3 \right)}^{3}}=-27\]
Hence, option (d) is the correct answer.

Note: One may note that when we have to calculate any odd power of any number which is negative then we will get a negative number in the result and if we have to calculate any even power of any number which may be either positive or negative, we will get a positive number in the result. In the above question, we were provided with a negative number whose power was odd and that is why we got a negative number as our answer.