Let ‘*’ and ‘^’ be two binary operations such that a*b=a2 b and a ^ b = 2a+b. Find (2*3) ^ (6*7)

The correct option is 276.

To find the value of the expression $(2*3)^(6*7)$, we need to apply the definitions of the two binary operations step-by-step.

The binary operations are defined as follows:
1. $a*b=a^{2}b$
2. $a^b=2^{a+b}$ (or $2^a+b$ or $2a+b$. Let us verify: if $a^b=2a+b$ as written in the text: $2*3=2^2\cdot 3=4\cdot 3=12$, and $6*7=6^2\cdot 7=36\cdot 7=252$. Then $12^252=2\left(12\right)+252=24+252=276$. This matches the correct answer perfectly!)

Now, let us calculate the individual parts of the expression:

Step 1: Calculate $2*3$
Using the definition $a*b=a^{2}b$, we substitute $a=2$ and $b=3$:
$2*3=2^2\times 3$
$2*3=4\times 3=12$

Step 2: Calculate $6*7$
Using the definition $a*b=a^{2}b$, we substitute $a=6$ and $b=7$:
$6*7=6^2\times 7$
$6*7=36\times 7=252$

Step 3: Calculate the final expression $(2*3)^(6*7)$
We substitute the values we computed in Step 1 and Step 2 into the expression:
$(2*3)^(6*7)=12^252$

Now, using the definition $a^b=2a+b$ where $a=12$ and $b=252$:
$12^252=2\left(12\right)+252$
$12^252=24+252$
$12^252=276$

Thus, the final evaluated value is indeed 276.