Question

The binary operation \[*:R\times R\to R\] is defined as \[a*b=2a+b\]. Using the definition find the value of \[(2*3)*4\].

Answer 1

Hint: In this question, We are given a binary operation \[*:R\times R\to R\] which is defined by
\[a*b=2a+b\] 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 \[(2*3)*4\].

Complete step-by-step answer:
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 \[(2*3)*4=18\].

Note: In this problem, while calculate the value of \[(a*b)=(2*3)\] we have to keep in mind the operation \[*\] is not a simple multiplication between the real numbers \[a\] and \[b\]. Rather \[*\] is a binary operation which is defined by \[a*b=2a+b\]. We have to use the same definition which is given otherwise the answer would be wrong.