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.
6a^4+162a/3a^4-9a^3+27a^2
Get used to using parentheses online, ok?
(6a^4+162a) / (3a^4-9a^3+27a^2)
(6a)(a^3+27) / (3a^2)(a^2-3a+9)
Now think back on how to factor x^3 ± y^3
and you will see some factors drop out.
To reduce the rational expression to its lowest terms, we need to simplify the numerator and the denominator as much as possible and then cancel out any common factors.
Let's start by factoring out any common factors from the numerator and denominator:
Numerator: 6a^4 + 162a = 6a(a^3 + 27)
Denominator: 3a^4 - 9a^3 + 27a^2 = 3a^2(a^2 - 3a + 9)
Now, let's cancel out any common factors between the numerator and the denominator:
6a(a^3 + 27) / [3a^2(a^2 - 3a + 9)]
There are no common factors left, so this expression is already in its lowest terms.
As for the restrictions on the variable, we need to consider any values that would make the denominator zero since division by zero is undefined. In this case, the denominator can be zero when a^2 - 3a + 9 = 0. However, this quadratic equation does not have any real solutions, so there are no restrictions on the variable a.
Therefore, the reduced rational expression in its lowest terms is 6a(a^3 + 27) / [3a^2(a^2 - 3a + 9)].