Question

simplify.
\\(\dfrac{24u^2(u + 6)^2}{18u^5(u + 6)(3u + 7)}\\)
you may leave the numerator and denominator of your answer in factored form.

Explanation:

Step1: Simplify the coefficients

First, simplify the numerical coefficients 24 and 18. The greatest common divisor of 24 and 18 is 6. So, divide both by 6: $\frac{24\div6}{18\div6} = \frac{4}{3}$.

Step2: Simplify the \(u\)-terms

For the \(u\)-terms, we have \(u^2\) in the numerator and \(u^5\) in the denominator. Using the rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\), we get \(\frac{u^2}{u^5}=u^{2-5}=u^{-3}=\frac{1}{u^3}\).

Step3: Simplify the \((u + 6)\)-terms

For the \((u + 6)\)-terms, we have \((u + 6)^2\) in the numerator and \((u + 6)\) in the denominator. Using the rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\), we get \(\frac{(u + 6)^2}{(u + 6)}=(u + 6)^{2-1}=u + 6\).

Step4: Combine the simplified terms

Now, combine all the simplified parts. The numerator parts are 4, \(u + 6\) and the denominator parts are 3, \(u^3\), \((3u + 7)\). So the simplified fraction is \(\frac{4(u + 6)}{3u^3(3u + 7)}\).

Answer:

\(\frac{4(u + 6)}{3u^3(3u + 7)}\)