Question

The LCM of \[{{4}^{5}},{{4}^{-81}},{{4}^{12}},{{4}^{7}}\]
(a) \[{{4}^{-81}}\]
(b) \[{{4}^{5}}\]
(c) \[{{4}^{12}}\]
(d) \[{{4}^{7}}\]

Answer 1

Hint: In this question, we first need to write the given terms in the form of \[\dfrac{p}{q}\]. Then we need to find the least common multiple of all the numerators and highest common factor of all the denominators. Now, on dividing the least common multiple of numerators with the highest common factor of the denominators gives the LCM of the given numbers.

Complete step by step solution:
FACTOR: A number which can divide a given number exactly, is called a factor of that number.
MULTIPLE: A number which is divisible by a given number, is called multiple of that number.
HIGHEST COMMON FACTOR(HCF):
HCF of two or more numbers is the greatest number, which divides all the given numbers exactly.
 LEAST COMMON MULTIPLE(LCM):
The least number which is exactly divisible by two or more given numbers is called LCM of those numbers.
LCM of Fractions:
LCM of the fraction numbers, after expressing them in their lowest from is given by
\[\Rightarrow \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}\]
Now, the given numbers in the question are
\[\Rightarrow {{4}^{5}},{{4}^{-81}},{{4}^{12}},{{4}^{7}}\]
Let us now express each of these numbers as fractions
\[\Rightarrow \dfrac{{{4}^{5}}}{1},\dfrac{1}{{{4}^{81}}},\dfrac{{{4}^{12}}}{1},\dfrac{{{4}^{7}}}{1}\]
Now, let us find the least common multiple of the numerators
\[\Rightarrow LCM\left( {{4}^{5}},1,{{4}^{12}},{{4}^{7}} \right)\]
Now, the least common multiple of all these numbers is
\[\therefore LCM\left( {{4}^{5}},1,{{4}^{12}},{{4}^{7}} \right)={{4}^{12}}\]
Let us now find the highest common factor of all the denominators in the above numbers expressed as fractions
\[\Rightarrow HCF\left( 1,{{4}^{81}},1,1 \right)\]
Now, the common factor of all these numbers is
\[\therefore HCF\left( 1,{{4}^{81}},1,1 \right)=1\]
Now, the LCM of the numbers can be found by using the formula
\[\Rightarrow \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}\]
Now, on substituting the respective values we get,
\[\Rightarrow \dfrac{{{4}^{12}}}{1}\]
Now, on further simplification we get,
\[\Rightarrow {{4}^{12}}\]
Hence, the correct option is (c).

Note: Instead of finding the least common multiple of numerators and highest common factor of denominators separately we can directly find them in the formula of LCM of fractions and simplify further.
It is important to note that we need to find the number which is the common multiple to all the numbers while finding the LCM and the number which is a common factor while finding the HCF. Because if we do them in the opposite way then the result changes completely.