Question

given \\(\frac{8^5}{8^2} = \frac{1}{8^{(5 + 2)}} = \frac{1}{8^7}\\)

Explanation:

Step1: Apply exponent division rule

For \(\frac{a^m}{a^n}\), the rule is \(a^{m - n}\). So \(\frac{8^5}{8^2}=8^{5 - 2}=8^3\)? Wait, no, wait the given step is transforming to \(\frac{1}{8^{(5 + 2)}}\). Wait, maybe it's a negative exponent rule. Recall that \(a^{-n}=\frac{1}{a^n}\) and \(\frac{a^m}{a^n}=a^{m - n}\). So \(\frac{8^5}{8^2}=8^{5-2}\)? But the given step is \(\frac{1}{8^{(5 + 2)}}\). Wait, maybe there's a mistake in the problem's step? Wait no, let's check again. Wait, maybe the original problem is \(\frac{8^2}{8^5}\)? Wait the given is \(\frac{8^5}{8^2}\) equals \(\frac{1}{8^{(5 + 2)}}\)? Wait no, that's incorrect. Wait, no, let's re - express. If we have \(\frac{8^5}{8^2}\), using \(\frac{a^m}{a^n}=a^{m - n}\), it should be \(8^{5 - 2}=8^3\). But the given step is \(\frac{1}{8^{(5+2)}}\), which would be \(8^{-(5 + 2)}=8^{-7}\), and \(\frac{8^5}{8^2}=8^{5-2}=8^3\), which is not equal to \(8^{-7}\). Wait, maybe the original fraction is \(\frac{8^2}{8^5}\). Let's assume that maybe there's a typo, and it's \(\frac{8^2}{8^5}\). Then \(\frac{8^2}{8^5}=8^{2-5}=8^{-3}\)? No, the next step is \(\frac{1}{8^{(5 + 2)}}=\frac{1}{8^7}\). Wait, \(5+2 = 7\), so if we have \(\frac{8^m}{8^n}=\frac{1}{8^{m + n}}\), that would imply \(m - n=-(m + n)\), \(m - n=-m - n\), \(2m = 0\), \(m = 0\), which is not the case here. Wait, maybe the problem is using a wrong rule. But let's follow the given steps. The first step: \(\frac{8^5}{8^2}\), then the second step is \(\frac{1}{8^{(5 + 2)}}\), using the rule that \(\frac{a^m}{a^n}=\frac{1}{a^{n - m}}\) when \(n>m\). Wait, if \(m = 5\) and \(n = 2\), no, \(n

First, the given expression is \(\frac{8^5}{8^2}\). Then, they transform it to \(\frac{1}{8^{(5 + 2)}}\). Let's check the exponent operation. If we use the negative exponent rule: \(a^{-n}=\frac{1}{a^n}\), and \(\frac{a^m}{a^n}=a^{m - n}\). So if \(m = 5\) and \(n = 2\), \(\frac{8^5}{8^2}=8^{5-2}=8^3\). But the given step is \(\frac{1}{8^{5 + 2}}\), which is \(8^{-(5 + 2)}=8^{-7}\). These are not equal, so there's a mistake in the problem's step. But assuming that the problem is correct as per its own presentation, let's see the arithmetic in the exponent addition: \(5+2 = 7\), so \(\frac{1}{8^7}\).

Step1: Analyze the exponent operation in the denominator

In the expression \(\frac{1}{8^{(5 + 2)}}\), we first calculate the exponent in the denominator. The exponent is \(5+2\).
\(5 + 2=7\)

Step2: Substitute the exponent value

Substitute \(7\) into the expression \(\frac{1}{8^{(5 + 2)}}\), we get \(\frac{1}{8^7}\)

Answer:

The final result of the given (though the first step of transformation from \(\frac{8^5}{8^2}\) to \(\frac{1}{8^{5 + 2}}\) is arithmetically incorrect as per the standard exponent division rule \(\frac{a^m}{a^n}=a^{m - n}\)) but following the given steps, the result is \(\frac{1}{8^7}\) (or \(8^{-7}\)).