Let '*' and '^' be two binary operations such that a * b = a$^2$b and a ^ b = 2a + b. Find (-4 * 2) ^ (7 * -1).

To solve this problem, we need to evaluate the expression step-by-step using the definitions provided for the binary operations.

1. Analyze the definitions

• Operation (*): $a * b = a^2b$
• Operation (^): $a \wedge b = 2a + b$

We need to find the value of: $(-4 * 2) \wedge (7 * -1)$

2. Step-by-step calculation

Step 1: Solve the first bracket $(-4 * 2)$ Using the rule $a * b = a^2b$: $(-4 * 2) = (-4)^2 \times 2$ $(-4 * 2) = 16 \times 2$ $(-4 * 2) = 32$

Step 2: Solve the second bracket $(7 * -1)$ Using the rule $a * b = a^2b$: $(7 * -1) = (7)^2 \times (-1)$ $(7 * -1) = 49 \times (-1)$ $(7 * -1) = -49$

Step 3: Combine the results using the (^) operation Now we have $(32) \wedge (-49)$. Using the rule $a \wedge b = 2a + b$: $32 \wedge -49 = 2(32) + (-49)$ $32 \wedge -49 = 64 - 49$ $32 \wedge -49 = 15$

3. Conclusion

Following the order of operations and the definitions provided, the result is 15. This matches option D.

Exam Tip: When dealing with binary operations, always treat the brackets as separate mini-problems. Solve the contents of the brackets first before applying the final operator. Also, be very careful with negative signs when squaring numbers (e.g., $(-4)^2$ is $+16$, not $-16$).