Question

simplify. express your answer using positive exponents. \\(\frac{8s^{0}}{4s}\\)

Explanation:

Step1: Apply zero exponent rule

Recall that any non - zero number to the power of 0 is 1, so \(s^{0}=1\) (assuming \(s
eq0\)). Then the numerator \(8s^{0}\) becomes \(8\times1 = 8\).

Step2: Simplify the fraction of coefficients and use exponent rule for variables

For the coefficients, we have \(\frac{8}{4}=2\). For the variables, we use the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\). Here, for the variable \(s\), we have \(\frac{s^{0}}{s}=\frac{1}{s^{1}}\) (since \(s^{0} = 1\)) or using the exponent rule \(s^{0-1}=s^{-1}=\frac{1}{s}\). But we need to combine the coefficient and the variable part. So \(\frac{8s^{0}}{4s}=\frac{8\times1}{4s}=\frac{8}{4s}\). Simplifying \(\frac{8}{4s}\), we get \(\frac{2}{s}\)? Wait, no, wait. Wait, let's do it step by step with exponents.

Wait, \(\frac{8s^{0}}{4s}=\frac{8}{4}\times\frac{s^{0}}{s}\). \(\frac{8}{4} = 2\), and \(\frac{s^{0}}{s}=s^{0 - 1}=s^{-1}=\frac{1}{s}\). But we need positive exponents. Wait, no, maybe I made a mistake. Wait, \(s^{0}=1\), so the expression is \(\frac{8\times1}{4s}=\frac{8}{4s}=\frac{2}{s}\)? Wait, no, that can't be. Wait, no, let's re - express:

\(\frac{8s^{0}}{4s}=\frac{8}{4}\times s^{0 - 1}\) (using the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\) for \(a = s\), \(m = 0\), \(n = 1\)) and \(\frac{8}{4}=2\). So \(2\times s^{-1}\). But we need positive exponents, and \(s^{-1}=\frac{1}{s}\), so \(2\times\frac{1}{s}=\frac{2}{s}\)? Wait, but let's check again. Wait, \(s^{0}=1\), so numerator is 8, denominator is 4s. So \(\frac{8}{4s}=\frac{2}{s}\). Yes, that's correct. Wait, but let's verify with \(s = 2\). Original expression: \(\frac{8\times(2)^{0}}{4\times2}=\frac{8\times1}{8}=1\). \(\frac{2}{2}=1\). Oh, right! Wait, if \(s = 2\), \(\frac{2}{s}=\frac{2}{2}=1\), and the original expression is \(\frac{8\times1}{8}=1\). So that works. Wait, but let's do it with exponents properly.

The zero - exponent rule: \(a^{0}=1\) for \(a
eq0\). So \(s^{0}=1\) (assuming \(s
eq0\)). Then \(\frac{8s^{0}}{4s}=\frac{8\times1}{4s}=\frac{8}{4s}\). Simplify the fraction \(\frac{8}{4}=2\), so we have \(\frac{2}{s}\). Alternatively, using exponent rules for the variable: \(\frac{s^{0}}{s}=s^{0 - 1}=s^{-1}\), and \(\frac{8}{4}=2\), so \(2\times s^{-1}\). Since \(s^{-1}=\frac{1}{s}\), then \(2\times\frac{1}{s}=\frac{2}{s}\).

Answer:

\(\frac{2}{s}\)