Multiply the expression \[3ab\times \left( 5{{a}^{2}}+4{{b}^{2}} \right)\] .

Hint: For solving these problems, we need to have a clear understanding of the rules of multiplication and indices. By employing the distributive law, and the rules of multiplication of indices, we can easily get the expression for the above problem.

Complete step-by-step solution:
In Maths, Algebra is one of the important branches. The concept of algebra is used to find the unknown variables or unknown quantity. The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants. Algebraic expression is an expression that is built by the combination of integer constants and variables. For example, \[4xy+9\] , in this expression x and y are variables whereas $4$ and $9$ are constants. The value of an algebraic expression changes according to the value chosen for the variables of the expressions.
When a number (let a) is multiplied by itself “n” times. For example, ${{a}^{n}}$ , here, ‘a’ is called the base and ‘n’ is known as the index of the power. Basically, it is an exponential expression. One of the basic rules of the multiplication of indices is ${{a}^{m}}\times {{a}^{n}}={{a}^{mn}}$ .
Distributive property of multiplication over the addition of algebraic expression is:
\[x\times \left( y+z \right)=\left( x\times y \right)+\left( x\times z \right)\]
Employing these laws, we may now write down the given expression as,
\[\begin{align}
  & \Rightarrow 3ab\times \left( 5{{a}^{2}}+4{{b}^{2}} \right) \\
 & \Rightarrow \left( 3ab\times 5{{a}^{2}} \right)+\left( 3ab\times 4{{b}^{2}} \right) \\
\end{align}\]
Now as we know, ${{a}^{m}}\times {{a}^{n}}={{a}^{mn}}$ , therefore, the expression results in
\[\Rightarrow 15{{a}^{3}}b+12a{{b}^{3}}\]
Thus, multiplication of \[3ab\times \left( 5{{a}^{2}}+4{{b}^{2}} \right)\] gives \[15{{a}^{3}}b+12a{{b}^{3}}\] .

Note: These types of problems are pretty easy to solve, but one needs to be careful otherwise small misjudgements can lead to a totally different answer. We need to carefully employ the properties of multiplication of indices to get to the correct answer and not get confused between the properties as well.