Answer

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**Hint:**In this problem we have given a fraction and asked to simplify it. We can observe that the given fraction is in the form of $\dfrac{a+b}{c}$, so we can write it as $\dfrac{a}{c}+\dfrac{b}{c}$. Now we will consider each fraction separately. In the first fraction we can observe that the imaginary number $i$ in the denominator. To rationalize it we are going to multiply and divide the fraction with $i$. Now we will use the formula ${{i}^{2}}=-1$ and simplify the fraction by cancelling the common factors that the both numerator and denominator have. Now coming to the second fraction, for this fraction also we will simplify it by cancelling the common factors in numerator and denominator. Now we will add the values of both the fractions to get the required result.

**Complete step-by-step solution:**

Given that, $\dfrac{4+9i}{12i}$.

The above fraction is in the form of $\dfrac{a+b}{c}$, so we are going to write it as $\dfrac{a}{c}+\dfrac{b}{c}$, then we will get

$\Rightarrow \dfrac{4+9i}{12i}=\dfrac{4}{12i}+\dfrac{9i}{12i}$

We can observe two fractions $\dfrac{4}{12i}$, $\dfrac{9i}{12i}$ in the above equation.

Considering the fraction $\dfrac{4}{12i}$. We have the imaginary number $i$ in the denominator, so we are going to multiply and divide the fraction with $i$, then we will get

$\dfrac{4}{12i}=\dfrac{4}{12i}\times \dfrac{i}{i}$

Multiplying the numerator with numerator and denominator with denominator, then we will have

$\Rightarrow \dfrac{4}{12i}=\dfrac{4i}{12{{i}^{2}}}$

We know that ${{i}^{2}}=-1$, substituting this value in the above equation, then we will get

$\Rightarrow \dfrac{4}{12i}=\dfrac{4i}{-12}$

Cancelling the common factor $4$ in both numerator and denominator, then we will get

$\Rightarrow \dfrac{4}{12i}=-\dfrac{1}{3}i$.

Now considering the fraction $\dfrac{9i}{12i}$. We can observe that the imaginary number $i$ is in both numerator and denominator, so we are going to cancelling the imaginary number along with the common factor which is $3$ in both numerator and denominator, then we will get

$\Rightarrow \dfrac{9i}{12i}=\dfrac{3}{4}$

Now the value of $\dfrac{4+9i}{12i}$ will be

$\Rightarrow \dfrac{4+9i}{12i}=\dfrac{3}{4}-\dfrac{1}{3}i$

**Hence the simplified value of $\dfrac{4+9i}{12i}$ is $\dfrac{3}{4}-\dfrac{1}{3}i$.**

**Note:**You can also directly multiply and divide the given fraction with the imaginary number $i$ to simplify the given value. But when you go with this you need to use the distribution law of multiplication in the numerator.

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