Question

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

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 $(2*3)*4$.

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.