Final answer:
The number 2.57x10^4 is exactly 10^5 times larger than 2.57x10^-1, due to the properties of powers of ten where the exponents are subtracted when dividing, and negative exponents indicate division by that power of ten.
Step-by-step explanation:
The relationship between 2.57×104 and 2.57×10−1 can be determined by understanding how the powers of ten operate. To make a comparison between these two numbers, we recognize that 2.57×104 is the larger number because it has a positive exponent, indicating that the decimal point has been moved to the right, increasing the value substantially. In contrast, 2.57×10−1 has a negative exponent, meaning the decimal point has been moved to the left, making it a fraction of the original number.
When comparing these two values directly, we can calculate that 2.57×104 is exactly 2.57×104 ÷ 2.57×10−1 = 104 ÷ 10−1 = 104+1 = 105 times larger than 2.57×10−1.