Question

simplify.\\(\dfrac{\dfrac{5x^3}{3x + 21}}{\dfrac{7x}{x + 7}}\\)

Explanation:

Step1: Rewrite as multiplication

Dividing by a fraction is multiplying by its reciprocal. So, \(\frac{\frac{5x^3}{3x + 21}}{\frac{7x}{x + 7}}=\frac{5x^3}{3x + 21}\times\frac{x + 7}{7x}\)

Step2: Factor \(3x + 21\)

Factor out 3 from \(3x + 21\), we get \(3(x + 7)\). So the expression becomes \(\frac{5x^3}{3(x + 7)}\times\frac{x + 7}{7x}\)

Step3: Cancel common factors

Cancel out \((x + 7)\) from numerator and denominator. Also, cancel \(x\) from \(x^3\) (leaving \(x^2\)) and 7, 3 remain. So we have \(\frac{5x^2}{3\times7}=\frac{5x^2}{21}\) (Wait, no, wait. Wait, let's redo step 3. Wait, \(\frac{5x^3}{3(x + 7)}\times\frac{x + 7}{7x}\). Cancel \(x + 7\), then \(x^3/x = x^2\), so \(5x^2/(3\times7)=\frac{5x^2}{21}\)? Wait, no, wait, original step 1: Wait, the original complex fraction: the numerator of the big fraction is \(\frac{5x^3}{3x + 21}\) and the denominator is \(\frac{7x}{x + 7}\). So dividing the two is \(\frac{5x^3}{3x + 21}\div\frac{7x}{x + 7}=\frac{5x^3}{3(x + 7)}\times\frac{x + 7}{7x}\). Then cancel \(x + 7\), cancel \(x\) (from \(x^3\) to \(x^2\)), so \(5x^2/(3\times7)=\frac{5x^2}{21}\)? Wait, no, wait, 3 and 7? Wait, no, 3 is from \(3(x + 7)\) and 7 is from \(7x\). So yes, \(5x^3\times(x + 7)\) over \(3(x + 7)\times7x\). Then cancel \(x + 7\), \(x\) (so \(x^3\) becomes \(x^2\)), so \(5x^2/(3\times7)=\frac{5x^2}{21}\). Wait, but let's check again. Wait, maybe I made a mistake in step 1. Wait, the original problem is \(\frac{\frac{5x^3}{3x + 21}}{\frac{7x}{x + 7}}\). So that's equal to \(\frac{5x^3}{3x + 21}\times\frac{x + 7}{7x}\). Then factor \(3x + 21 = 3(x + 7)\). So substitute: \(\frac{5x^3}{3(x + 7)}\times\frac{x + 7}{7x}\). Then cancel \(x + 7\), cancel \(x\) (so \(x^3/x = x^2\)), so numerator: \(5x^2\), denominator: \(3\times7 = 21\). So the simplified form is \(\frac{5x^2}{21}\)? Wait, no, wait, 5x^3 times (x +7) is 5x^3(x +7), denominator is 3(x +7) times 7x. So cancel (x +7), cancel x (x^3 /x = x^2), so 5x^2 / (3*7) = 5x^2/21. Yes, that's correct.

Answer:

\(\frac{5x^2}{21}\)