Question

astronomy the distance from earth to the moon is about 22^4 miles. the distance from earth to neptune is about 22^7 miles. which distance is the greater distance, and about how many times greater is it?

Explanation:

Step1: Compare the exponents

We know that for positive - base \(a\) (\(a> 1\)) in the exponential form \(a^m\) and \(a^n\), if \(m>n\), then \(a^m > a^n\). Here \(a = 22\), \(m = 7\), \(n = 4\). Since \(7>4\), \(22^{7}>22^{4}\). The distance from Earth to Neptune (\(22^{7}\) miles) is greater.

Step2: Calculate the ratio

To find out how many times greater \(22^{7}\) is than \(22^{4}\), we use the rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\). So \(\frac{22^{7}}{22^{4}}=22^{7 - 4}=22^{3}\). Calculate \(22^{3}=22\times22\times22=(20 + 2)\times(20 + 2)\times(20+2)=(400 + 80+4)\times(20 + 2)=(484)\times(20 + 2)=9680+968 = 10648\).

Answer:

The distance from Earth to Neptune is greater. It is about \(22^{3}=10648\) times greater than the distance from Earth to the moon.