Question

problem 6 part b
here is the table from the previous screen.
how many digits will be in the decimal for the fraction \\(\frac{1}{256}\\)?

fraction	decimal	number of digits in decimal
\\(\frac{1}{4}\\)	0.25	2
\\(\frac{1}{8}\\)	0.125	3
\\(\frac{1}{16}\\)	0.0025	4
\\(\frac{1}{32}\\)	0.03125	5

Explanation:

Step1: Analyze the pattern

Observe the given fractions: $\frac{1}{2}=2^1$, decimal digits = 1; $\frac{1}{4}=2^2$, decimal digits = 2; $\frac{1}{8}=2^3$, decimal digits = 3; $\frac{1}{16}=2^4$, decimal digits = 4; $\frac{1}{32}=2^5$, decimal digits = 5. The pattern is that for $\frac{1}{2^n}$, the number of decimal digits is $n$.

Step2: Express 256 as a power of 2

We know that $256 = 2^8$ (since $2^8 = 256$). So the fraction $\frac{1}{256}=\frac{1}{2^8}$.

Step3: Determine the number of digits

Using the pattern from Step 1, for $\frac{1}{2^n}$, the number of decimal digits is $n$. Here, $n = 8$, so the number of digits in the decimal for $\frac{1}{256}$ is 8.

Answer:

8