Question

How do you simply \[{{y}^{9}}\div {{y}^{5}}\]?

Answer 1

Hint: We have to simplify the given expression. We will first multiply and divide by \[{{y}^{-5}}\]. The numerator has \[{{y}^{9}}\], so when multiplied with \[{{y}^{-5}}\], the powers get added up as the base is same (which is \[y\]). And we get \[{{y}^{4}}\]. In the denominator we have \[{{y}^{5}}\], which when multiplied to \[{{y}^{-5}}\], the power gets added up and we get \[{{y}^{0}}\] which is equal to 1. So, we have the simplified form of the expression.

Complete step by step solution:
According to the given question, we have been given an expression which we have to simplify as much as possible. The expression which we have is,
\[{{y}^{9}}\div {{y}^{5}}\], we can also write it as,
\[\Rightarrow \dfrac{{{y}^{9}}}{{{y}^{5}}}\]
We will begin by multiplying and dividing by \[{{y}^{-5}}\]. Since, we are multiplying and dividing by the same entity the expression is still the same. We get,
\[\Rightarrow \dfrac{{{y}^{9}}}{{{y}^{5}}}\times \dfrac{{{y}^{-5}}}{{{y}^{-5}}}\]
In the numerator, we have \[{{y}^{9}}\] and \[{{y}^{-5}}\], since they both have the same base, their powers will get added. That is,
\[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
And in the denominator, we have \[{{y}^{5}}\] and \[{{y}^{-5}}\], since they both have the same base, their powers will get added as well. So we have,
\[\Rightarrow \dfrac{{{y}^{9}}^{-5}}{{{y}^{5}}^{-5}}\]
Solving the powers of the above expression, we have,
\[\Rightarrow \dfrac{{{y}^{4}}}{{{y}^{0}}}\]
As we know, \[{{y}^{0}}=1\], so we get the result as,
\[\Rightarrow \dfrac{{{y}^{4}}}{1}\]
\[\Rightarrow {{y}^{4}}\]

Therefore, the simplified form is \[{{y}^{4}}\]

Note: We can also simplify the given expression using the property we know, that is, \[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\].
So, we have,
\[{{y}^{9}}\div {{y}^{5}}\]
\[\Rightarrow \dfrac{{{y}^{9}}}{{{y}^{5}}}\]
\[\Rightarrow {{y}^{9-5}}\]
\[\Rightarrow {{y}^{4}}\]
Therefore, the simplified form is \[{{y}^{4}}\].