Use the given digits so each expression has the greatest possible value. Express the answer as a mixed number in simplest form. (a) $$ 4,5,6 $$

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To find the greatest possible value using the digits $$4$$, $$5$$, and $$6$$ in an expression, we can consider different operations such as addition, subtraction, multiplication, and division.

Let's explore the possible combinations:

1. **Addition**: 
   $$
   4 + 5 + 6 = 15
   $$

2. **Multiplication**: 
   $$
   4 \times 5 \times 6 = 120
   $$

3. **Division**: 
   - If we divide, we can try different combinations, but generally, division will yield a smaller result compared to multiplication.
   - For example:
     $$
     \frac{6}{4} + 5 = 1.5 + 5 = 6.5
     $$
   - Other combinations will yield even smaller results.

4. **Mixed Operations**: 
   - We can also try combinations of operations:
     $$
     6 \times (5 - 4) = 6 \times 1 = 6
     $$
     $$
     (6 + 5) \times 4 = 11 \times 4 = 44
     $$

From the calculations above, the maximum value is obtained through multiplication:

$$
4 \times 5 \times 6 = 120
$$

Since the problem asks for the answer as a mixed number in simplest form, we can express $$120$$ as:

$$
120 = 120 \frac{0}{1}
$$

Thus, the final answer is:

$$
\boxed{120}
$$
The greatest possible value using the digits 4, 5, and 6 is 120.

Use the given digits so each expression has the greatest possible value. Express the answer as a mixed number in simplest form. (a) \( 4,5,6 \)

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