Question

Evaluate the value of ${\left( {104} \right)^3}$ using the suitable identities.

Answer 1

Hint: In the given question, we have to evaluate the cube of a number given to us in the problem itself with the help of an algebraic identity. We must remember the algebraic identity for the whole cube of a binomial expression: ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$. The algebraic identity ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$ is used to evaluate the cube of a binomial expression involving the sum of two terms.

Complete step-by-step answer:
Given question requires us to find the value of a cube of $104$.
We will use the algebraic identity ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$ for expanding the expression and simplifying it further.
So, to calculate the cube of $104$, we have to first divide the number $104$ into two parts such that the following calculation of the cube of the number becomes easier.
So, we know that $104 = 100 + 4$. So, we can divide $104$ into the two numbers $100$ and $4$. So, we now substitute the values of the two parts into the algebraic identity that we are supposed to use.
Hence, we have, ${\left( {104} \right)^3} = {\left( {100 + 4} \right)^3}$
Now, we expand the left side of the equation using the algebraic identity to evaluate the cube of a binomial expression involving the sum of two terms. So, we get,
$ \Rightarrow {\left( {104} \right)^3} = {100^3} + {4^3} + 3\left( {100} \right)\left( 4 \right)\left( {100 + 4} \right)$
$ \Rightarrow {\left( {104} \right)^3} = 1000000 + 64 + 1200 \times 104$
$ \Rightarrow {\left( {104} \right)^3} = 1000000 + 64 + 124800$
$ \Rightarrow {\left( {104} \right)^3} = 1124864$
So, the value of ${\left( {104} \right)^3}$ calculated using the algebraic identity ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$ is $1124864$.
So, the correct answer is “1124864”.

Note: We can also verify the answer of the given question by calculating the cube of $104$ simply. We can calculate the cube of $104$ by multiplying the number three times with itself because $a \times a \times a = {a^3}$. Hence, ${104^3} = \left( {104 \times 104} \right) \times 104 = 10816 \times 104 = 1124864$. Therefore, the cube of the number $104$ is $1124864$.