If [tex]$^{(\ \textless \ em\ \textgreater \ )}$[/tex] is an operation defined for two integers [tex]$a$[/tex] and [tex]$b$[/tex] such that [tex]$a^\ \textless \ /em\ \textgreater \ b=(a \times a)+(b \times b)-(a \times b)$[/tex], find the value of [tex]$a^* b$[/tex] when [tex]$a=-3$[/tex] and [tex]$b=2$[/tex].



Answer :

To find the value of [tex]\( a^ b \)[/tex] using the operation defined as [tex]\( a^ b = (a \times a) + (b \times b) - (a \times b) \)[/tex], we need to substitute the given values [tex]\( a = -3 \)[/tex] and [tex]\( b = 2 \)[/tex] into the operation and simplify step by step.

1. Substitute [tex]\( a = -3 \)[/tex] and [tex]\( b = 2 \)[/tex] into the expression:
[tex]\[ (-3)^* 2 = (-3 \times -3) + (2 \times 2) - (-3 \times 2) \][/tex]

2. Calculate each of the individual multiplications:
- [tex]\((-3 \times -3)\)[/tex]
- [tex]\((2 \times 2)\)[/tex]
- [tex]\((-3 \times 2)\)[/tex]

Substituting in the values:
[tex]\[ (-3 \times -3) = 9 \][/tex]
[tex]\[ (2 \times 2) = 4 \][/tex]
[tex]\[ (-3 \times 2) = -6 \][/tex]

3. Substitute these results back into the original expression:
[tex]\[ (-3)^* 2 = 9 + 4 - (-6) \][/tex]

4. Simplify the expression inside the parentheses:
[tex]\[ 9 + 4 - (-6) = 9 + 4 + 6 \][/tex]

5. Complete the addition:
[tex]\[ 9 + 4 + 6 = 19 \][/tex]

Therefore, the value of [tex]\( a^* b \)[/tex] when [tex]\( a = -3 \)[/tex] and [tex]\( b = 2 \)[/tex] is:
[tex]\[ a^* b = 19 \][/tex]