10 ### Simplifying Expressions: Lowest Terms**Question:**Express in lowest terms:\[\frac{27a^3 - 1}{9a^2 - 6a + 1}\]**Options:**- ( ) 1- ( ) \(\frac{1}{2}\)- ( ) \(3a - 1\)- ( ) \(\frac{9a^2 + 3a + 1}{3a - 1}\)**Explanation:**To simplify the given expression, we need to factorize both the numerator and the denominator. First, let's factorize the numerator \(27a^3 - 1\):\[27a^3 - 1 = (3a)^3 - 1^3\]We recognize this as a difference of cubes, which can be factored using the formula:\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]Applying this to our expression:\[27a^3 - 1 = (3a - 1)(9a^2 + 3a + 1)\]Next, we factorize the denominator \(9a^2 - 6a + 1\):The denominator is already in its simplest form and can be factored as:\[9a^2 - 6a + 1 = (3a - 1)^2\]Now, substituting the factored forms back into the fraction, we get:\[\frac{(3a - 1)(9a^2 + 3a + 1)}{(3a - 1)^2}\]We can cancel out the common factor \((3a - 1)\):\[\frac{9a^2 + 3a + 1}{3a - 1}\]Thus, the expression in its lowest terms is:\[\boxed{\frac{9a^2 + 3a + 1}{3a - 1}}\]**Graphical Representation:**This question requires simplifying a polynomial fraction to its lowest term. There are no graphs or diagrams involved in the solution of this particular problem.**Navigation:**- [Next Question]- [Ask for Help]**Copyright:**© 2007, 2009,

Name: 10 ### Simplifying Expressions: Lowest Terms**Question:**Express in lo
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: 10 ### Simplifying Expressions: Lowest Terms**Question:**Express in lowest terms:\[\frac{27a^3 - 1}{9a^2 - 6a + 1}\]**Options:**- ( ) 1- ( ) \(\frac{1}{2}\)- ( ) \(3a - 1\)- ( ) \(\frac{9a^2 + 3a + 1}{3a - 1}\)**Explanation:**To simplify the given ex