To convert the denominator into a power of 10, we multiply the numerator and denominator by 5³ so that the exponents of 2 and 5 both become 6. This makes the denominator equal to (2 · 5)^6 = 10^6.

After how many decimal place will the decimal expansion of the rational number 359 upon 2 ki power 6 into 5 ki power 3

Answer: The decimal expansion terminates after 6 decimal places.

Step-by-step solution

Step 1: Check coprimality of numerator and denominator

The given rational number is $\frac{359}{2^6 \cdot 5^3}$ . Since the prime factorization of the denominator contains only the prime factors $2$ and $5$ , and $359$ is not divisible by $2$ or $5$ , the fraction is already in its simplest irreducible form.

Step 2: Make powers of 2 and 5 equal in the denominator

To convert the denominator into a power of $10$ , we multiply the numerator and denominator by $5^3$ so that the exponents of $2$ and $5$ both become $6$ . This makes the denominator equal to $(2 \cdot 5)^6 = 10^6$ .

Step 3: Determine the number of decimal places

Since the denominator is $10^6$ , dividing by $10^6$ shifts the decimal point $6$ places to the left. In general, for a rational number in simplest form $\frac{p}{2^n \cdot 5^m}$ , the decimal expansion terminates after $\max(n, m)$ decimal places. Here, $\max(6, 3) = 6$ .

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