Question

If y = 3, then \[{{y}^{3}}\left( {{y}^{3}}-y \right)=\]
(a) 300
(b) 459
(c) 648
(d) 999
(e) 1099

Answer 1

Hint: To find the value of the term given in the question, we will put the value of y = 3 in \[{{y}^{3}}\left( {{y}^{3}}-y \right)\]. Then according to the formula: \[{{y}^{3}}=y\times y\times y\], we will calculate \[{{y}^{3}}\] and put it in the term given in the above, and then we will open the bracket and multiply \[{{y}^{3}}\text{ with }{{y}^{3}}-y\] to obtain the answer.

Complete step-by-step answer:
In this question, we are given that the value of y is 3 and we have to find the value of \[{{y}^{3}}\left( {{y}^{3}}-y \right)\] with the help of this information. To do this, we will assume that the value of \[{{y}^{3}}\left( {{y}^{3}}-y \right)\] is ‘t’. Thus, we have the following equation:
\[t={{y}^{3}}\left( {{y}^{3}}-y \right).....\left( i \right)\]
We can see that the right-hand side of the equation is a multiplication of two terms of which one term is \[{{y}^{3}}\] and the other term is \[\left( {{y}^{3}}-y \right)\]. So, first, we will calculate the value of \[{{y}^{3}}\].
Calculation of \[{{y}^{3}}\]:
We know that \[{{y}^{3}}\] can also be written as,
\[{{y}^{3}}=y\times y\times y\]
Now, we know that the value of y given is equal to 3. After putting this value in the above equation, we get,
\[{{y}^{3}}=3\times 3\times 3\]
\[{{y}^{3}}=27.....\left( ii \right)\]
Calculation of \[\left( {{y}^{3}}-y \right)\]:
We have to calculate the value \[\left( {{y}^{3}}-y \right)\]. The value of \[{{y}^{3}}\] is calculated by using the above part. So, we will subtract y from the value of this equation. We know that the value of y given in the question is equal to 3. Thus, we get,
\[{{y}^{3}}-y=27-3\]
\[{{y}^{3}}-y=24....\left( iii \right)\]
Now, we will put the value of \[{{y}^{3}}\text{ and }\left( {{y}^{3}}-y \right)\] from equation (ii) and (iii) into (i). After doing this, we will get,
\[\begin{align}
  & t=27\left( 24 \right) \\
 & t=27\times 24 \\
 & t=648.....\left( iv \right) \\
\end{align}\]
From (i) and (iv), we get,
\[{{y}^{3}}\left( {{y}^{3}}-y \right)=648\]
Hence, the option (c) is the right answer.

Note: The value of the term \[{{y}^{3}}\left( {{y}^{3}}-y \right)\] can also be calculated as shown:
We can write \[{{y}^{3}}\left( {{y}^{3}}-y \right)\] as \[{{y}^{3}}\left( y \right)\left( {{y}^{2}}-1 \right)\]. Thus, we get,
\[t={{y}^{4}}\left( {{y}^{2}}-1 \right)\]
Now, we will find the value of \[{{y}^{4}}\]. We get,
\[{{3}^{4}}=81\]
Therefore,
\[t=81\left( {{y}^{2}}-1 \right)\]
Now, we will use another identity
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]
Therefore,
\[t=81\left( y-1 \right)\left( y+1 \right)=81\left( 2 \right)\left( 4 \right)=648\]