Question

Simplify the expression. $$\frac{x}{x+6} \div \frac{x+1}{x+6}$$

Short answer

\(\frac{x^2+6x}{x^2+7x+6}\)

Step-by-step solution

Rewrite the Division as a Multiplication

We rewrite the expression \(\frac{x}{x+6} \div \frac{x+1}{x+6}\) as multiplication by the reciprocal, which gives us \(\frac{x}{x+6} \times \frac{x+6}{x+1}\).

Multiply the Fractions

We then multiply the two fractions by each other. The multiplication of fractions involves multiplying the numerators (top parts of the fractions) with each other, and the denominators (bottom parts) with each other. So, we will have \(\frac{x \times (x+6)}{(x+6) \times (x+1)}\). In simplified form, the result is \(\frac{x^2+6x}{x^2+7x+6}\).

Simplify the Resulting Fraction

Upon inspection, it doesn't seem like the fraction can be simplified any further, so the final answer remains \(\frac{x^2+6x}{x^2+7x+6}\).