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\[a*b=2a+b\] where \[a,b\in R\]. Now is order to calculate the value of \[(2*3)*4\], we will first find the value of \[(2*3)\] by substituting \[a=2\] and \[b=3\] in \[a*b=2a+b\]. We will then substitute the value of \[(2*3)\] in \[(2*3)*4\]. Suppose we get that \[(2*3)=x\], then we will substitute this value in \[(2*3)*4\] and then we will get \[(2*3)*4=x*4\] . Then again we will calculate the binary operation between the value of \[(2*3)\] and 4 nu substituting \[a=x\] and \[b=4\] in \[a*b=2a+b\] to get the desired value of \[(2*3)*4\].

We are given with a binary operation \[*:R\times R\to R\] which is defined by

\[a*b=2a+b\] where \[a,b\in R\]

Now in order to find the value of \[(2*3)*4\], we will first calculate the value of \[(2*3)\].

First suppose that \[a=2\] and \[b=3\].

We will now calculate the value of \[(a*b)=(2*3)\] by using the given definition of \[*\].

\[\begin{align}

& (a*b)=(2*3) \\

& =2\left( 2 \right)+3 \\

& =4+3 \\

& =7

\end{align}\]

Thus we have that the value of \[(2*3)=7\].

Now in order to calculate \[(2*3)*4\], we will first substitute the value \[(2*3)=7\] in \[(2*3)*4\].

On substituting the value of \[(2*3)=7\] , we get

\[(2*3)*4=7*4\]

Now we will calculate the value of \[7*4\] using the definition of \[*\].

For that let us suppose \[a=7\] and \[b=4\].

Then we have

\[\begin{align}

& (a*b)=(7*4) \\

& =2\left( 7 \right)+4 \\

& =14+4 \\

& =18

\end{align}\]

Hence we get that the value of \[(2*3)*4=18\].