Question

find the quotient.
\\(\frac{34a^3b^4 + 54a^4b^3 + 74ab}{2ab}\\)
enter the correct answer.

Explanation:

Step1: Split the fraction into three terms

We can split the given fraction \(\frac{34a^{3}b^{4}+54a^{4}b^{3}+74ab}{2ab}\) into three separate fractions: \(\frac{34a^{3}b^{4}}{2ab}+\frac{54a^{4}b^{3}}{2ab}+\frac{74ab}{2ab}\)

Step2: Simplify each term

• For the first term \(\frac{34a^{3}b^{4}}{2ab}\), divide the coefficients and use the quotient rule for exponents (\(x^m\div x^n = x^{m - n}\)):

The coefficient \(34\div2 = 17\), for \(a\) we have \(a^{3}\div a=a^{3 - 1}=a^{2}\), for \(b\) we have \(b^{4}\div b=b^{4 - 1}=b^{3}\). So the first term simplifies to \(17a^{2}b^{3}\)

• For the second term \(\frac{54a^{4}b^{3}}{2ab}\), divide the coefficients \(54\div2 = 27\), for \(a\) we have \(a^{4}\div a=a^{4 - 1}=a^{3}\), for \(b\) we have \(b^{3}\div b=b^{3 - 1}=b^{2}\). So the second term simplifies to \(27a^{3}b^{2}\)
• For the third term \(\frac{74ab}{2ab}\), divide the coefficients \(74\div2=37\), and \(ab\div ab = 1\). So the third term simplifies to \(37\)

Answer:

\(17a^{2}b^{3}+27a^{3}b^{2}+37\)