Question

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

Answer 1

Hint: In this question, we are going to simplify the equation by using rules of exponents but at the same time we have to ensure that the exponents should be positive. So, we will use quotient rules to cancel the powers and then, we will leave our answer there.

Formula used: In this question, we are going to use the following formula of rules for exponents:
$\dfrac{{{a^n}}}{{{a^m}}} = {a^{n - m}}$
$\dfrac{{{a^n}}}{{{b^n}}} = {\left( {\dfrac{a}{b}} \right)^n}$

Complete step by step solution:
In this question, first we are going to simplify the given equation by using quotient rules.
The given equation is $\dfrac{{{m^4}}}{{2{m^4}}}$
We can also simplify the question and write it as follows:
$\dfrac{{{m^4}}}{{2{m^4}}} = \dfrac{1}{2}\left( {\dfrac{{{m^4}}}{{{m^4}}}} \right)$
Here $m$ and $2$ is called the base element and $4$ is called the exponent or index.
In this $m$ is multiplied by itself $4$ times.
Here the base element in the numerator and the denominator are the same.
Now we are going to flip the denominator to the numerator.
$\Rightarrow$$\dfrac{{{m^4}}}{{2{m^4}}} = \dfrac{1}{2}{m^{4 - 4}}$
$\Rightarrow$$\dfrac{{{m^4}}}{{2{m^4}}} = \dfrac{1}{2}{m^0}$
$\Rightarrow$$\dfrac{{{m^4}}}{{2{m^4}}} = \dfrac{1}{2}\left( 1 \right)$
(Since any nonzero expression to the zero power is one, that is anything to the power zero is one)
$\Rightarrow$$\dfrac{{{m^4}}}{{2{m^4}}} = \dfrac{1}{2}$
A positive exponent tells us how many times to multiply a base number.
$\Rightarrow$$\dfrac{{{m^4}}}{{2{m^4}}} = 0.5$
Finally we can get the required result as a positive exponent.

Therefore, the positive, simplified exponent of $\dfrac{{{m^4}}}{{2{m^4}}}$ is $\dfrac{1}{2}.$

Note: There are various rules of exponent. Some of them are listed below
Product rules, quotient rules, power rules, negative exponents, zero rules, one rule, minus one rule, derivative rule, integral rule.
In this question we have used the quotient rule only.
Also, in this we are asked to write the result only as a positive exponent otherwise we could have also stated our answer as $ = {2^{ - 1}}$ .