Question

simplify.
\\(\frac{6k^2m - 12k^3m^2 + 9m^3}{2km^2}\\)
\\(\bigcirc\\) \\(\frac{3k}{m} - 6k^2 + \frac{9m}{2k}\\)
\\(\bigcirc\\) \\(3km - 6k^2 + 9m2k\\)
\\(\bigcirc\\) \\(\frac{3k3}{m} - \frac{6k^5}{m^4} + \frac{9m^5}{2k}\\)
\\(\bigcirc\\) \\(\frac{3k}{m} + 6k^2 - \frac{9m}{2k}\\)

Explanation:

Step1: Split the fraction

We can split the given fraction \(\frac{6k^{2}m - 12k^{3}m^{2}+9m^{3}}{2km^{2}}\) into three separate fractions:
\[
\frac{6k^{2}m}{2km^{2}}-\frac{12k^{3}m^{2}}{2km^{2}}+\frac{9m^{3}}{2km^{2}}
\]

Step2: Simplify each fraction

• For the first fraction \(\frac{6k^{2}m}{2km^{2}}\):

Using the rule of exponents \(a^{m}\div a^{n}=a^{m - n}\) and dividing the coefficients, we have:
The coefficient \(6\div2 = 3\), for \(k\): \(k^{2}\div k=k^{2 - 1}=k\), for \(m\): \(m\div m^{2}=m^{1-2}=m^{- 1}=\frac{1}{m}\)
So \(\frac{6k^{2}m}{2km^{2}}=\frac{3k}{m}\)

• For the second fraction \(\frac{12k^{3}m^{2}}{2km^{2}}\):

The coefficient \(12\div2 = 6\), for \(k\): \(k^{3}\div k=k^{3 - 1}=k^{2}\), for \(m\): \(m^{2}\div m^{2}=m^{2-2}=m^{0} = 1\)
So \(\frac{12k^{3}m^{2}}{2km^{2}}=6k^{2}\), and since it's a subtraction in the original expression, we have \(-\frac{12k^{3}m^{2}}{2km^{2}}=- 6k^{2}\)

• For the third fraction \(\frac{9m^{3}}{2km^{2}}\):

The coefficient remains \(9\div2\) (but we can write it as is for now), for \(k\): \(k^{0}\div k=k^{-1}=\frac{1}{k}\), for \(m\): \(m^{3}\div m^{2}=m^{3 - 2}=m\)
So \(\frac{9m^{3}}{2km^{2}}=\frac{9m}{2k}\)

Combining these simplified fractions, we get \(\frac{3k}{m}-6k^{2}+\frac{9m}{2k}\)

Answer:

A. \(\frac{3k}{m}-6k^{2}+\frac{9m}{2k}\)