Question

If \[{{2}^{m}}+{{2}^{1+m}}=24\], then determine the value of \[m\] which will satisfy the given condition.
(a) 0
(b) \[\dfrac{1}{3}\]
(c) 3
(d) 6

Answer 1

Hint: In this question, We are given with the equation \[{{2}^{m}}+{{2}^{1+m}}=24\]. We can take common \[{{2}^{m}}\] from the left hand side of the equation and simplify the equation. We will then try to get powers of 2 on both sides of the equation. Then we can equate the powers of 2 from both the sides to get the desired value for \[m\].

Complete step-by-step answer:
We are given that \[{{2}^{m}}+{{2}^{1+m}}=24\].
Consider the right hand side of the given equation. It can be rewritten as
\[{{2}^{m}}+{{2}^{m}}\times 2=24\]
Since the value \[{{2}^{m}}\] can be taken common from both the terms in the right hand side of the equation \[{{2}^{m}}+{{2}^{1+m}}=24\].
Thus on taking \[{{2}^{m}}\] common on the right side of the above equation we have
 \[{{2}^{m}}\left( 1+2 \right)=24\]
Now on simplifying the above equation we get
\[{{2}^{m}}\left( 3 \right)=24\]
On dividing both right hand side and left hand side of the above equation by 3, we get
\[\dfrac{{{2}^{m}}\left( 3 \right)}{3}=\dfrac{24}{3}\]
\[\Rightarrow {{2}^{m}}=8\]
Now we know that the nature number 8 can be expressed as
\[8={{2}^{3}}\]
There we have
\[{{2}^{m}}={{2}^{3}}\]
Now on equating the powers of 2 in the above equation, we will get
\[m=3\]
Hence the required value of \[m\] is equal to 3.

So, the correct answer is “Option C”.

Note: In this problem, try to take the highest common divisor of both the terms of the right hand side of the equation \[{{2}^{m}}+{{2}^{1+m}}=24\] which is \[{{2}^{m}}\] rather than taking the 2 as common and then solve the equation in order to simplify the problem. If 2 is taken as a common factor from the right side of the equation \[{{2}^{m}}+{{2}^{1+m}}=24\], then it would be difficult to proceed further.