4) 34 Simplify.Express your answer as a single fraction in simplest form. 3q-(q)/(9q+1) square

To simplify the expression \(3q - \frac{q}{9q+1}\), we need to combine the terms into a single fraction. Here are the steps:<br /><br />1. **Identify a common denominator**: The common denominator for the terms \(3q\) and \(\frac{q}{9q+1}\) is \(9q + 1\).<br /><br />2. **Rewrite each term with the common denominator**:<br /> - For the term \(3q\), we need to express it as a fraction with the denominator \(9q + 1\):<br /> \[<br /> 3q = \frac{3q(9q + 1)}{9q + 1}<br /> \]<br /> - The term \(\frac{q}{9q + 1}\) already has the common denominator.<br /><br />3. **Combine the fractions**:<br /> \[<br /> 3q - \frac{q}{9q + 1} = \frac{3q(9q + 1)}{9q + 1} - \frac{q}{9q + 1}<br /> \]<br /><br />4. **Combine the numerators over the common denominator**:<br /> \[<br /> \frac{3q(9q + 1) - q}{9q + 1}<br /> \]<br /><br />5. **Simplify the numerator**:<br /> \[<br /> 3q(9q + 1) - q = 27q^2 + 3q - q = 27q^2 + 2q<br /> \]<br /><br />6. **Write the final simplified expression**:<br /> \[<br /> \frac{27q^2 + 2q}{9q + 1}<br /> \]<br /><br />Therefore, the simplified form of the expression \(3q - \frac{q}{9q+1}\) is:<br />\[<br />\boxed{\frac{27q^2 + 2q}{9q + 1}}<br />\]