ISTANCE A long-distance runner runs \( 2^{5} \) miles one week and \( 2^{7} \) miles the next week How many times farther did he run in the second week than in the first week?

In the first week, the runner covered \( 2^{5} = 32 \) miles. In the second week, he ran \( 2^{7} = 128 \) miles. To find out how many times farther he ran in the second week than in the first week, you can divide the distance of the second week by the distance of the first week: \[ \frac{128}{32} = 4 \] So, the runner ran 4 times farther in the second week than in the first week! In case you're curious about the concept of exponents, it's worth noting that \( 2^{m} \) can be expressed as \( 2^{n} \times 2^{k} \) where \( m = n + k \). So here, \( 2^{7} = 2^{5} \times 2^{2} \), meaning he ran \( 2^{2} \) or 4 times more in the second week! For further understanding, exploring exponential growth can be enlightening! It's fascinating how exponents work not just in running but in various aspects of life, from finance (think compound interest) to population growth. Look into "The Joy of Exponents" for a deeper dive!