Question

(\frac{6a^{6}b^{-5}c^{5}}{4b^{-3}c^{-5}})

Explanation:

Step1: Simplify the coefficient

Simplify the fraction of the coefficients \( \frac{6}{4} \). We can divide both the numerator and the denominator by their greatest common divisor, which is 2. So \( \frac{6}{4}=\frac{6\div2}{4\div2}=\frac{3}{2} \).

Step2: Simplify the \(a\) term

For the \(a\) term, since there is only \(a^6\) in the numerator and no \(a\) term in the denominator, the \(a\) term remains \(a^6\).

Step3: Simplify the \(b\) term

Using the rule of exponents \( \frac{x^m}{x^n}=x^{m - n} \) for the \(b\) terms. The exponent of \(b\) in the numerator is \(-5\) and in the denominator is \(-3\). So \( b^{-5-(-3)}=b^{-5 + 3}=b^{-2}=\frac{1}{b^2} \).

Step4: Simplify the \(c\) term

Using the rule of exponents \( \frac{x^m}{x^n}=x^{m - n} \) for the \(c\) terms. The exponent of \(c\) in the numerator is \(5\) and in the denominator is \(-5\). So \( c^{5-(-5)}=c^{5 + 5}=c^{10} \).

Step5: Combine all the simplified terms

Multiply the simplified coefficient, \(a\) term, \(b\) term, and \(c\) term together. We get \( \frac{3}{2}\times a^6\times\frac{1}{b^2}\times c^{10}=\frac{3a^6c^{10}}{2b^2} \).

Answer:

\( \frac{3a^6c^{10}}{2b^2} \)