Final answer:
To simplify the expression, we use properties of exponents to combine like bases and apply zero exponent rule. The final simplification of the exponential expression is 1 ÷ 256, which does not match any given option (a. 2, b. 1, c. 1/2) as they do not represent the correct result.
Step-by-step explanation:
To simplify the exponential expression 3⁻³· 2⁻³· 6³ ÷ (4²)², let's break it down step-by-step using the properties of exponents.
First, we can rewrite the expression by splitting the 6³ as (3· 2)³ which gives us:
3⁻³ · 2⁻³ · (3· 2)³ ÷ (4²)²
According to the property of exponents for multiplication, we can add the exponents of like bases. This yields:
(3⁻³ · 3³) · (2⁻³ · 2³) ÷ (4²)²
= 3·(3⁻³+3) · 2·(2⁻³+3) ÷ (4´)
Since the exponents of the same base are added, 3⁻³+3 = 0 and 2⁻³+3 = 0, which simplifies to:
3·(0) · 2·(0) ÷ (4´)
Anything raised to the zero power is 1. So this simplifies further to:
1 ÷ (4´)
The expression (4²)² is cubing of exponentials, which means we multiply the exponents: 2² = 4. So the denominator becomes 4´ or 16². Thus, our expression becomes:
1 ÷ 16²
Simplifying the denominator:
1 ÷ 256
In conclusion, the correct answer, 1 ÷ 256, is a fraction less than 1, hence the correct match would be with option c. 1/2. However, there is an issue with the given options because none of them accurately represent the result of 1 ÷ 256.