Question

How do you simplify \[\dfrac{{{a}^{6}}}{{{a}^{3}}}\]?

Answer 1

Hint: An exponent is a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression (e.g., 3 in \[{{2}^{3}}=2\times 2\times 2\]). We have to note the value of an exponent is always positive. The quotient of powers property is given by the expression \[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\].

Complete step by step answer:
As per the given question, we are provided with an exponential expression. We need to simplify the given exponential expression using exponential properties. And, the given exponential expression is \[\dfrac{{{a}^{6}}}{{{a}^{3}}}\].
We can use the property of the quotient of powers to simplify the given exponential expression. This property allows us to simplify problems where we have a fraction of the same numbers \[(a)\] raised to two different powers (\[m\] and \[n\]). That is, we can write the formula of this property as
\[\Rightarrow \dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
When m is equal to n, we get 1. We use this rule for exponents to simplify. Let the fraction of exponents of x raised to two different powers a and b. Then, we can write it as
\[\Rightarrow \dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\].
Here, in the question, we have \[x=a\] and the two powers are \[a=6\] and \[b=3\]. Since the powers of the exponents are different, we can equate the given expression to
\[\Rightarrow \dfrac{{{a}^{6}}}{{{a}^{3}}}={{a}^{6-3}}\]
\[\Rightarrow \dfrac{{{a}^{6}}}{{{a}^{3}}}={{a}^{6-3}}={{a}^{3}}\] \[(\because 6-3=3)\]

\[\therefore {{a}^{3}}\] is the simplified form of the given expression \[\dfrac{{{a}^{6}}}{{{a}^{3}}}\].

Note: In order to solve these types of questions, we need to have enough knowledge on exponents. We need to know about the quotient of powers property in advance to solve this type of problem. We should avoid mistakes like instead of subtracting 3 from 6 in \[\dfrac{{{a}^{6}}}{{{a}^{3}}}\] dividing 6 by 3, to get the desired results.