Question

What is the LCM of $({a^3} + {b^3})$ and $({a^4} - {b^4})$?

Answer 1

Hint: In this question, we are given two numbers in terms of a and b, and we have been asked to find their LCM. Start by expanding the numbers using their properties. Then take the highest power of all of their factors and multiply them together. The resultant term(s) will be the answer..

Complete step by step answer:
We are given two numbers in terms of a and b.
We will use the formulas of square and cubes and then we will expand them.
This will give us their factors. Using these factors, we will find the LCM.
Using the formula to find the factors-
$ \Rightarrow {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$
$ \Rightarrow {a^4} - {b^4} = \left( {{a^2} + {b^2}} \right)\left( {{a^2} - {b^2}} \right)$
$ \Rightarrow {a^4} - {b^4} = \left( {{a^2} + {b^2}} \right)\left( {a + b} \right)\left( {a - b} \right)$
Now, before finding LCM, let us know what is LCM.
The Least Common Multiple (LCM) of certain numbers is the smallest number that is the multiple of all the numbers. If we observe, this number is bigger than all the given numbers of which LCM is to be found.
Let us understand by an example: LCM of 34 and 40. At first, we will find their factors:
$34 = 2 \times 17$
$40 = {2^3} \times 5$
To find LCM, we take the highest values of all the factors. For example, the highest power of 2 is 3.
LCM = ${2^3} \times 5 \times 17$
LCM = $680$
Hence, LCM of 34 and 40 is 680.
Now, let us move towards our question. We have found their factors. Now, just like we took the highest value of all the factors in the above example, we will do the same here.
LCM = $\left( {a + b} \right)\left( {a - b} \right)\left( {{a^2} + {b^2}} \right)\left( {{a^2} - ab + {b^2}} \right)$
LCM = $\left( {{a^3} + {b^3}} \right)\left( {{a^2} + {b^2}} \right)\left( {a - b} \right)$

Therefore, the LCM of $({a^3} + {b^3})$ and $({a^4} - {b^4})$ is $\left( {{a^3} + {b^3}} \right)\left( {{a^2} + {b^2}} \right)\left( {a - b} \right)$.

Note: To find out HCF and LCM, we can use the following shortcut method:
To find LCM, take the highest powers of the factors.
To find HCF, take the lowest powers of the factors.
This method will simplify the process.
For example: Let us take 2 numbers x and y. Their factors are given below. Find their HCF and LCM.
$x = {2^3} \times {3^2} \times {5^2} \times {7^5} \times 11 \times {13^4}$
$y = {2^2} \times {3^3} \times 5 \times {7^3} \times {13^6}$
HCF (take lowest powers) = ${2^2} \times {3^2} \times 5 \times {7^3} \times {13^4}$
LCM (take highest powers) = ${2^3} \times {3^3} \times {5^2} \times {7^5} \times 11 \times {13^6}$
Therefore, using these rules will make your work easier.