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]
