Question

# If the operation $\oplus$ is defined by $a\oplus b={{a}^{2}}+{{b}^{2}}$ for all real numbers $a$ and $b$, then $(2\oplus 3)\oplus 4$ .

Hint: We are given $a\oplus b={{a}^{2}}+{{b}^{2}}$ and we have to find $(2\oplus 3)\oplus 4$ . For that, first take $(2\oplus 3)$ and apply $a\oplus b={{a}^{2}}+{{b}^{2}}$. After that, the number we get from $(2\oplus 3)$ then apply it with $4$ . Try it, you will get the answer.

The basic operations of mathematics- addition, subtraction, division and multiplication are performed on two operands. Even when we try to add three numbers, we add two of them and then add the third number to the result of the two numbers. Thus, the basic mathematical operations are performed on two numbers and are known as binary operations (the word binary means two). In this section, we will discuss binary operations performed on a set.
According to digital electronics and mathematics, a binary number is defined as a number that is expressed in the binary system or base 2 numeral system. It describes numeric values by two separate symbols; basically 1 (one) and 0 (zero). The base-2 system is the positional notation with 2 as a radix. The binary system is applied internally by almost all latest computers and computer-based devices because of its direct implementation in electronic circuits using logic gates. Every digit is referred to as a bit.
We take the set of numbers on which the binary operations are performed as $X$ . The operations (addition, subtraction, division, multiplication, etc.) can be generalized as a binary operation is performed on two elements (say $a$ and $b$) from set $X$ . The result of the operation on$a$and $b$ is another element from the same set $X$ .
Thus, the binary operation can be defined as an operation * which is performed on a set $A$. The function is given by *: $A*A\to A$ . So the operation * performed on operands $a$ and $b$is denoted by $a*b$ .
We are given, $a\oplus b={{a}^{2}}+{{b}^{2}}$ .
Now, $(2\oplus 3)\oplus 4$ ,
taking $(2\oplus 3)={{2}^{2}}+{{3}^{3}}=4+9=13$
Now, $13\oplus 4={{13}^{2}}+{{4}^{2}}=169+16=185$ .
So we get , $(2\oplus 3)\oplus 4=185$ .

Note: Read the question carefully. Don’t confuse yourself. Your concept regarding binary operation should be clear. Also, take care that while simplifying no terms are missed. Do not make any silly mistakes. While solving, take care that no signs are missed.