Question

How do you simplify the Mathematical expression \[\dfrac{{{x^8}}}{{{x^4}}}\]using properties?

Answer 1

Hint:
According to the question, we can solve it by using two properties. We can use the quotient of power property and the power of a power property. First, we will use the properties, to simplify it and then solve it.

Complete step by step solution:
We can solve the above equations in two ways:
The first method through which we will solve the question is by using the quotient of powers property. The formula of this property is:
\[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]
So, all we need to do is put the values into this formula. Here, \[a = x;\,m = 8;\,n = 4\].
\[ \Rightarrow \dfrac{{{x^8}}}{{{x^4}}} = {x^{8 - 4}}\]
Here you can see how the power of 8 in the numerator is getting reduced by the presence of the power 4 in the denominator. After that:
\[ \Rightarrow \dfrac{{{x^8}}}{{{x^4}}} = {x^4}\]
So, the result is \[{x^4}\].
The other method is by using the power of a power property of multiplication. The formula is:
\[{({a^m})^n} = {a^{m \cdot n}}\]
Here, \[a = x;\,m = 8;\,n = 4\]. According to this formula, we need to convert the term \[{x^8}\] in the form of \[{x^4}\]:
\[ \Rightarrow \dfrac{{{x^8}}}{{{x^4}}} = \dfrac{{{{({x^4})}^2}}}{{{x^4}}}\]
\[ \Rightarrow \dfrac{{{x^8}}}{{{x^4}}} = \dfrac{{({x^4})({x^4})}}{{{x^4}}}\]
Now, the similar terms from the numerator and the denominator will get cancel out, and we will get:
\[ \Rightarrow \dfrac{{{x^8}}}{{{x^4}}} = {x^4}\]

Therefore, our final answer is \[{x^4}\].

Note:
The quotient property which we used in the above solution says that if we are dividing two powers that are having the same base, then we can subtract the exponents. The power of a power property says that if we want to find a power of a term which is also having power, then we can multiply the exponents.