Question

Reduce the rational expression to lowest terms. If it is already in lowest terms, enter the expression in the answer box. Also, specify any restrictions on the variable.(a − 1)²(a - 5)/(a - 5)²(a - 1)Rational expression in lowest terms:Variable restrictions for the original expression: a

Answer 1

We have the following rational expression:

`((a-1)^2(a-5))/((a-5)^2(a-1))`

And we have to reduce the expression to the lowest terms.

1. To achieve that, we can see that:

• There are two common factors:

`(a-1)\text{ and }(a-5)`

2. Now, we know that:

`\begin{gathered} a^2=a*a \\ \\ \text{ And also, we have that:} \\ \\ (a)/(a)=(b)/(b)=1 \end{gathered}`

3. And we can apply the same rule for the given case as follows:

`\begin{gathered} ((a-1)^(2)(a-5))/((a-5)^(2)(a-1))\Rightarrow(a-1)^2=(a-1)(a-1),(a-5)^2=(a-5)(a-5) \\ \\ \text{ Then we have:} \\ \\ ((a-1)(a-1)(a-5))/((a-5)(a-5)(a-1)) \end{gathered}`

4. And we can rewrite the expression as follows:

`\begin{gathered} ((a-1)(a-1)(a-5))/((a-5)(a-5)(a-1))=((a-1))/((a-1))((a-5))/((a-5))((a-1))/((a-5))\Rightarrow((a-1))/((a-1))=1,((a-5))/((a-5))=1 \\ \\ \text{ Then we have:} \\ \\ ((a-1)(a-1)(a-5))/((a-5)(a-5)(a-1))=(1)(1)((a-1))/((a-5))=((a-1))/((a-5)) \\ \\ \end{gathered}`

Therefore, the rational expression of the lowest terms is:

`(a-1)/(a-5)`

5. We can determine that the variable restrictions for the original expression can be found as follows:

Therefore, in summary, we have that:

• The rational expression in the lowest terms is:

`(a-1)/(a-5)`

• The variable restrictions for the original expression:

`a\\e1,5`