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

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 $*$.

& (a*b)=(2*3) \\\ & =2\left( 2 \right)+3 \\\ & =4+3 \\\ & =7 \end{aligned}$$ Thus we have that the value of $$(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{aligned} & (a*b)=(7*4) \\\ & =2\left( 7 \right)+4 \\\ & =14+4 \\\ & =18 \end{aligned}$$ Hence we get that the value of $$(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.