This problem challenges us to evaluate a complex mathematical expression using two specially defined custom operators, ⊕ and ⊙. To solve it accurately, we need to understand the definitions of these operators and apply them correctly, following the standard order of operations.

Operators Definition and Functionality

First, let's clearly define how the given mathematical operators work:

• The operator ⊙ (read as "circle dot") is defined by the rule:
  \(\text{p } \odot \text{ q = p - q}\)
  This means that when you use the ⊙ operator, you subtract the second number (q) from the first number (p).
• The operator ⊕ (read as "circle plus") is defined by the rule:
  \(\text{p } \oplus \text{ q = p } \times \text{ q}\)
  This means that when you use the ⊕ operator, you multiply the first number (p) by the second number (q).

Expression Evaluation: Step-by-Step Process

We are asked to evaluate the following expression using these custom operators:

\((9 \odot (6 \oplus 7)) \odot (7 \oplus (6 \odot 5))\)

To evaluate this expression, we will follow the standard order of mathematical operations (PEMDAS/BODMAS), which dictates that operations inside parentheses should be performed first, from the innermost to the outermost.

Evaluating Inner Parentheses Operations

Step 1: Evaluate the innermost term \((6 \oplus 7)\)

According to the definition of the ⊕ operator (\(\text{p } \oplus \text{ q = p } \times \text{ q}\)):

\(6 \oplus 7 = 6 \times 7 = 42\)

Step 2: Evaluate the other innermost term \((6 \odot 5)\)

According to the definition of the ⊙ operator (\(\text{p } \odot \text{ q = p - q}\)):

\(6 \odot 5 = 6 - 5 = 1\)

Substituting and Evaluating Outer Parentheses

Now, we substitute the results from Step 1 and Step 2 back into the main expression. The expression now simplifies to:

\((9 \odot 42) \odot (7 \oplus 1)\)

Step 3: Evaluate the first outer parenthesis term \((9 \odot 42)\)

Using the definition of the ⊙ operator (subtraction):

\(9 \odot 42 = 9 - 42 = -33\)

Step 4: Evaluate the second outer parenthesis term \((7 \oplus 1)\)

Using the definition of the ⊕ operator (multiplication):

\(7 \oplus 1 = 7 \times 1 = 7\)

Final Evaluation of the Complete Expression

Substitute the results from Step 3 and Step 4 back into the simplified expression. The expression now becomes:

\((-33) \odot (7)\)

Step 5: Perform the final operation \((-33) \odot 7\)

Using the definition of the ⊙ operator (subtraction) one last time:

\(-33 \odot 7 = -33 - 7 = -40\)

Therefore, the final result of the given expression is \(-40\).