Final answer:
Negative exponents indicate a base in the denominator; thus, 3^-2x can be rewritten as (1 / 3^x)^2. Knowing that 3^x = 8, we substitute 8 for 3^x, and 3^-2x then equals (1 / 8)^2, which simplifies to 1 / 64.
Step-by-step explanation:
If 3x = 8, we want to find the value of 3-2x. To solve for 3-2x, we need to understand the properties of exponents. Negative exponents indicate that the base is in the denominator rather than the numerator. Therefore, 3-2x = 1 / 32x = (1 / 3x)2. Since we know that 3x = 8, we can substitute 8 in for 3x, giving us (1 / 8)2 as the final expression for 3-2x.
Applying this to the initial equation: 3x = 8 can be rewritten as x = log38. Since 3-2x = (1 / 3x)2, and substituting in the value for x, the equation becomes (1 / 3log38)2 = (1 / 8)2 = 1 / 64. Thus, 3-2x = 1 / 64.