Question

How do you simplify $ \dfrac{{2{m^{ - 4}}}}{{{{(2{m^{ - 4}})}^3}}} $ and write it using only positive exponents?

Answer 1

Hint: We know that the above given question is in exponential form. An exponent refers to the number of times a number is multiplied by itself. There is base and exponent or power in this type of equation. Here, in the given question $ (2m) $ is the base and the number $ - 4,3 $ are the exponential power. As we know that as per the property of exponent rule if there is $ \dfrac{{{a^m}}}{{{a^n}}} $ then it can be written as $ {a^{m - n}} $ . When we express a number in exponential form then we can say that it’s power has been raised by the exponent.

Complete step by step solution:
There is one basic exponential rule that is commonly used everywhere,
 $ {({a^m})^n} = {a^{m \cdot n}} $ .
We can simplify this by using the exponent rule
 $ \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} $ but if the higher power is in the denominator then we can write it as $ \dfrac{1}{{{a^{n - m}}}} $ . As in the given question we also have an inverse power, there is also another rule we have to apply which is
 $ {a^{ - 1}} = \dfrac{1}{a} $ .
To solve exponential equations with base, use the property of power of exponential functions. We will first break the denominator part by exponential rule i.e.
 $ \Rightarrow {(2{m^{ - 4}})^3} = {2^3} \times {m^{4 \times 3}} $ . It gives us $ {2^3}{m^{ - 12}} $ .
We will now put it in the given expression i.e.
 $ \Rightarrow \dfrac{{2{m^{ - 4}}}}{{{2^3}{m^{ - 12}}}} $ .
Now we will solve them by applying the above rule:
 $ \Rightarrow \dfrac{{{m^{ - 4 - ( - 12)}}}}{{{2^{3 - 1}}}} $ , since the power of $ 2 $ is greater in the denominator and the power of $ m $ is greater in the numerator.
Therefore it gives us $ \dfrac{{{m^{ - 4 + 12}}}}{{{2^2}}} = \dfrac{{{m^8}}}{4} $ .
Hence the required answer of the exponential form is $ \dfrac{{{m^8}}}{4} $ .
So, the correct answer is “ $ \dfrac{{{m^8}}}{4} $ .”.

Note: We know that exponential equations are equations in which variables occur as exponents. The formula applied before is true for all real values of $ m $ and $ n $ . We should solve this kind of problem by using the properties of exponents to simplify the problem. We have to keep in mind that if there is a negative value in the power or exponent then it will reverse the number .i.e. $ {m^{ - x}} $ will always be equal to $ \dfrac{1}{{{m^x}}} $ . We should know that the most commonly used exponential function base is the transcendental number which is denoted by $ e $ .