Question

Evaluate the following by using identities:
i) $98{}^3$
ii) ${103^3}$
iii) ${99^3}$
iv) ${1001^3}$

Answer 1

Hint: First observe that the given numbers are very near to some numbers which we know the cubes of. Similarly the difference is also so small that we can split the given terms in two numbers which we can easily cube. Then determine the proper identity that we can use to simplify the given terms.

Complete step-by-step answer:
First observe that all the terms are cubic terms so we have to use the terms in which we have cubes of some general terms.
Choosing a proper identity is the first important step before writing the solution.
We will list out both the identities that we will use to simplify the given terms.
The first identity is:
${\left( {a + b} \right)^3} = {a^3} + 3ab\left( {a + b} \right) + {b^3}$
The second identity is similar to the first one:
${\left( {a - b} \right)^3} = {a^3} - 3ab\left( {a - b} \right) - {b^3}$
We will use these two identities and solve the given problems one by one.

i) ${98^3}$
We will write $98 = 100 - 2$ and use the second identity to solve the problem.
Note that we know how to find the cubes of $100$ and $2$ .
We write the first step as:
${98^3} = {\left( {100 - 2} \right)^3}$
Using the expansion formula, we write,
${98^3} = {100^3} - 3\left( {100} \right)\left( 2 \right)\left( {100 - 2} \right) - {2^3}$
Now we know that ${100^3} = 1000000$ and ${2^3} = 8$ .
Substituting in the expansion we write:
${98^3} = 1000000 - 58800 - 8$
On simplifying we get:
${98^3} = 941192$

ii) ${103^3}$
We will write $103 = 100 + 3$ and use the second identity to solve the problem.
Note that we know how to find the cubes of $100$ and $3$ .
We write the first step as:
${103^3} = {\left( {100 + 3} \right)^3}$
Using the expansion formula, we write,
${103^3} = {100^3} + 3\left( {100} \right)\left( 3 \right)\left( {100 + 3} \right) + {3^3}$
Now we know that ${100^3} = 1000000$ and ${3^3} = 27$ .
Substituting in the expansion we write:
${103^3} = 1000000 + 92700 + 27$
On simplifying we get:
${103^3} = 1092727$

iii) ${99^3}$
We will write $99 = 100 - 1$ and use the second identity to solve the problem.
Note that we know how to find the cubes of $100$ and $1$ .
We write the first step as:
${99^3} = {\left( {100 - 1} \right)^3}$
Using the expansion formula, we write,
${99^3} = {100^3} - 3\left( {100} \right)\left( 1 \right)\left( {100 - 1} \right) - {1^3}$
Now we know that ${100^3} = 1000000$ and ${1^3} = 1$ .
Substituting in the expansion we write:
${99^3} = 1000000 - 29700 - 1$
On simplifying we get:
${99^3} = 970299$

iv) ${1001^3}$
We will write $1001 = 1000 + 1$ and use the second identity to solve the problem.
Note that we know how to find the cubes of $1000$ and $1$ .
We write the first step as:
${1001^3} = {\left( {1000 + 1} \right)^3}$
Using the expansion formula, we write,
${1001^3} = {1000^3} + 3\left( {1000} \right)\left( 1 \right)\left( {1000 + 1} \right) + {1^3}$
Now we know that ${1000^3} = 1000000000$ and ${1^3} = 1$ .
Substituting in the expansion we write:
${1001^3} = 1000000000 + 3003000 + 1$
On simplifying we get:
${1001^3} = 1003003001$

Note: First to identify the correct identity is the important step here. Then splitting up the terms so that the simplification can be done is another important point. Lastly, the calculations should be carried out precisely.