Question

simplify: \\(dfrac{dfrac{15n^{5}}{7p^{4}q^{2}}}{dfrac{5mn^{3}}{14p^{2}q}}\\)

Explanation:

Step1: Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

The expression is \(\frac{\frac{15n^{5}}{7p^{4}q^{2}}}{\frac{5mn^{3}}{14p^{2}q}}\), which is equivalent to \(\frac{15n^{5}}{7p^{4}q^{2}} \times \frac{14p^{2}q}{5mn^{3}}\).

Step2: Simplify the coefficients and variables separately.

For the coefficients: \(\frac{15\times14}{7\times5}=\frac{210}{35} = 6\).
For the \(n\) terms: \(\frac{n^{5}}{n^{3}}=n^{5 - 3}=n^{2}\).
For the \(p\) terms: \(\frac{p^{2}}{p^{4}}=p^{2 - 4}=p^{-2}=\frac{1}{p^{2}}\).
For the \(q\) terms: \(\frac{q}{q^{2}}=q^{1 - 2}=q^{-1}=\frac{1}{q}\).
For the \(m\) term: \(\frac{1}{m}\) (since there is no \(m\) in the numerator of the first fraction).

Step3: Combine all the simplified parts.

Multiply the simplified coefficients and variables together: \(6\times\frac{1}{m}\times n^{2}\times\frac{1}{p^{2}}\times\frac{1}{q}=\frac{6n^{2}}{mp^{2}q}\).

Answer:

\(\frac{6n^{2}}{mp^{2}q}\)