Mathematical Operator Definitions

The problem defines three unique mathematical operators: \(\boldsymbol{\text{∎}}\), \(\boldsymbol{\Delta}\), and \(\boldsymbol{\rightarrow}\). Understanding how each operator works is crucial to solving the given expression. Let's look at their definitions:

• Operator \(\boldsymbol{\text{∎}}\): For any two numbers \(a\) and \(b\), \(a \∎ b\) is defined as the difference between \(a\) and \(b\) divided by their sum.
  Mathematically, \(a \∎ b = \frac{{a - b}}{{a + b}}\).
• Operator \(\boldsymbol{\Delta}\): For any two numbers \(a\) and \(b\), \(a \Delta b\) is defined as the sum of \(a\) and \(b\) divided by their difference.
  Mathematically, \(a \Delta b = \frac{{a + b}}{{a - b}}\).
• Operator \(\boldsymbol{\rightarrow}\): For any two numbers \(a\) and \(b\), \(a \rightarrow b\) is defined as the product of \(a\) and \(b\).
  Mathematically, \(a \to b = ab\).

We need to find the value of the expression \((66 \∎ 6) \rightarrow (66 \Delta 6)\). We will solve this step-by-step by first evaluating the terms inside the parentheses.

Evaluating Operator Expression (66 ∎ 6)

First, let's calculate the value of \(66 \∎ 6\) using the definition of the \(\boldsymbol{\text{∎}}\) operator, where \(a = 66\) and \(b = 6\).

We have: $$66 \∎ 6 = \frac{{66 - 6}}{{66 + 6}}$$ $$66 \∎ 6 = \frac{{60}}{{72}}$$

To simplify the fraction, we can find the greatest common divisor (GCD) of 60 and 72. Both 60 and 72 are divisible by 12.

$$66 \∎ 6 = \frac{{60 \div 12}}{{72 \div 12}}$$ $$66 \∎ 6 = \frac{5}{6}$$ So, the value of the first part of the expression, \((66 \∎ 6)\), is \(\frac{5}{6}\).

Evaluating Operator Expression (66 \(\boldsymbol{\Delta}\) 6)

Next, let's calculate the value of \(66 \Delta 6\) using the definition of the \(\boldsymbol{\Delta}\) operator, where \(a = 66\) and \(b = 6\).

We have: $$66 \Delta 6 = \frac{{66 + 6}}{{66 - 6}}$$ $$66 \Delta 6 = \frac{{72}}{{60}}$$

To simplify this fraction, we again divide the numerator and the denominator by their GCD, which is 12.

$$66 \Delta 6 = \frac{{72 \div 12}}{{60 \div 12}}$$ $$66 \Delta 6 = \frac{6}{5}$$ So, the value of the second part of the expression, \((66 \Delta 6)\), is \(\frac{6}{5}\).

Final Operator Expression Calculation

Now we have the values for both parts of the main expression \((66 \∎ 6) \rightarrow (66 \Delta 6)\).

• \((66 \∎ 6) = \frac{5}{6}\)
• \((66 \Delta 6) = \frac{6}{5}\)

We need to apply the \(\boldsymbol{\rightarrow}\) operator to these two results. The definition of \(a \rightarrow b\) is \(ab\).

Let \(A = (66 \∎ 6) = \frac{5}{6}\) and \(B = (66 \Delta 6) = \frac{6}{5}\).

Then, the expression becomes \(A \rightarrow B = A \times B\).

$$ (66 \∎ 6) \rightarrow (66 \Delta 6) = \frac{5}{6} \times \frac{6}{5} $$

When multiplying these two fractions, the numerator of the first fraction cancels out with the denominator of the second fraction, and the denominator of the first fraction cancels out with the numerator of the second fraction.

$$ \frac{5}{6} \times \frac{6}{5} = \frac{5 \times 6}{6 \times 5} = \frac{30}{30} = 1 $$

Therefore, the final value of the expression \((66 \∎ 6) \rightarrow (66 \Delta 6)\) is \(\mathbf{1}\).