Question

simplify: \\(\frac{27k^{5}m^{8}}{(4k^{3})(9m^{2})}\\)

1. \\(\frac{27k^{2}m^{6}}{36}\\)
2. \\(\frac{3k^{8}m^{10}}{4}\\)
3. \\(\frac{27k^{8}m^{10}}{36}\\)
4. \\(\frac{3k^{2}m^{6}}{4}\\)

Explanation:

Step1: Simplify the constants and variables separately

First, simplify the constant terms: \( \frac{27}{4\times9} \), and then simplify the variable terms for \( k \) and \( m \) using the quotient rule of exponents \( \frac{a^m}{a^n}=a^{m - n} \).

For the constant part: \( \frac{27}{4\times9}=\frac{27}{36}=\frac{3}{4} \) (dividing numerator and denominator by 9).

For the \( k \) terms: \( \frac{k^5}{k^3}=k^{5 - 3}=k^2 \) (using \( \frac{a^m}{a^n}=a^{m - n} \)).

For the \( m \) terms: \( \frac{m^8}{m^2}=m^{8 - 2}=m^6 \) (using \( \frac{a^m}{a^n}=a^{m - n} \)).

Step2: Combine the simplified parts

Multiply the simplified constant, \( k \) term, and \( m \) term together: \( \frac{3}{4}\times k^2\times m^6=\frac{3k^2m^6}{4} \).

Answer:

1. \( \frac{3k^2m^6}{4} \)