Question

perform the operation and simplify.\frac{10r^{4}}{6n^{3}}\cdot\frac{42n^{4}}{34r^{5}}\frac{?n^{\quad}}{\quad r^{\quad}}

Explanation:

Step1: Multiply numerators and denominators

Multiply the numerators \(10r^4\) and \(42n^4\), and the denominators \(6n^3\) and \(34r^5\). So we get \(\frac{10r^4\times42n^4}{6n^3\times34r^5}\).

Step2: Simplify coefficients

Simplify the coefficients \(10\times42 = 420\) and \(6\times34 = 204\). Then \(\frac{420}{204}\) can be simplified by dividing numerator and denominator by 12, giving \(\frac{35}{17}\).

Step3: Simplify variables with exponents

For the \(n\) terms: using the rule \(a^m\div a^n=a^{m - n}\), we have \(n^4\div n^3=n^{4 - 3}=n^1\). For the \(r\) terms: \(r^4\div r^5=r^{4 - 5}=r^{-1}=\frac{1}{r^1}\).

Step4: Combine all simplified parts

Putting it all together, we have \(\frac{35n}{17r}\).

Answer:

\(\frac{35n}{17r}\) (So the number in the green box is 35, the exponent of \(n\) is 1, the number below \(n\) is 17, and the exponent of \(r\) is 1)