Question

The cube root of \[\sqrt[3]{-1331}\] is
(A) -11
(B) -21
(C) -31
(D) -19

Answer 1

Hint: First of all, get the factors of 1331. We can see that 1331 is divisible by 11, so one of its factors is 11. Now, we can see that -1331 can be written as \[11\times 121\] . We can see that 121 is also divisible by 11, so one of the factors of 121 is 11. Now, we can see that 1331 can be written as \[11\times 11\times 11\] . Now, using this transform \[\sqrt[3]{-1331}\] and replace -1 by \[\left( -1 \right)\left( -1 \right)\left( -1 \right)\] . Now, solve it further.

Complete step-by-step answer:
According to the question, it is given that our number is -1331 and we have to find the cube root of -1331.
So, we can say that we have to get the value of \[\sqrt[3]{-1331}\] .
First of all, we have to find the factors of -1331 and then only we can get the cube root of \[\sqrt[3]{-1331}\] ………………….(1)
We can see that -1 can be written as the triple product of -1.
\[-1=-1\times -1\times -1\] ………………….(2)
Therefore, -1331 can be written as \[-1\times -1\times -1\times 1331\].
Now, we need the factors of 1331.
We can see that 1331 is divisible by 11. In other words, we can say that 11 is a factor of the number 1331.
\[1331=11\times 121\] ……………………..(3)
Therefore, 1331 can be written as \[11\times 121\] .
Now, we need the factors of the number 121.
We can see that 121 is divisible by 11. So, 11 is also a factor of 121.
\[121=11\times 11\] ……………………(4)
Therefore, 121 can be written as \[11\times 11\] .
Using equation (2), equation (3), equation (4), -1331 can be written as,
\[-1331=\left( -1 \right)\left( -1 \right)\left( -1 \right)\left( 11 \right)\left( 11 \right)\left( 11 \right)\] ………………….(5)
We have to find the value of \[\sqrt[3]{-1331}\] ………………………….(6)
Now, putting the value of -1331 from equation (5) in equation (6), we get
\[\sqrt[3]{-1331}=\sqrt[3]{\left( -1 \right)\left( -1 \right)\left( -1 \right)\left( 11 \right)\left( 11 \right)\left( 11 \right)}=\left( -1 \right)\left( 11 \right)=-11\]
So, the value of the cube root of -1331 is -11, that is \[\sqrt[3]{-1331}=-11\] .
Therefore, the cube root of \[\sqrt[3]{-1331}\] is -11.
Hence, the correct option is (A).

Note: We can also solve questions by observing all the options and then we can figure out. We know that \[{{10}^{3}}=1000\] and \[{{20}^{3}}=4000\] . The number 1331 is between these two, so the cube root of 1331 should be between 10 and 20. We also know that if the unit of a number is 1, then the unit digit of the cube of that number will also be 1. Now, out of all the options given we have only option (A) which has the number lying between 10 and 20, and also has 1 at the unit place.