Question

simplify. assume all variables are positive.
\\(\frac{s^{\frac{5}{4}}}{s^{\frac{7}{4}}}\\)
write your answer in the form \\(a\\) or \\(\frac{a}{b}\\), where \\(a\\) and \\(b\\) are constants or variable expressions that have no variables in common. all exponents in your answer should be positive.

Explanation:

Step1: Apply exponent division rule

When dividing exponents with the same base, we subtract the exponents: \( \frac{s^{a}}{s^{b}} = s^{a - b} \). Here, \( a=\frac{5}{4} \) and \( b = \frac{7}{4} \), so we calculate \( \frac{5}{4}-\frac{7}{4} \).
\( \frac{5}{4}-\frac{7}{4}=\frac{5 - 7}{4}=\frac{-2}{4}=-\frac{1}{2} \)
So the expression becomes \( s^{-\frac{1}{2}} \).

Step2: Rewrite negative exponent

A negative exponent means the reciprocal with a positive exponent: \( s^{-n}=\frac{1}{s^{n}} \). So \( s^{-\frac{1}{2}}=\frac{1}{s^{\frac{1}{2}}} \), and \( s^{\frac{1}{2}}=\sqrt{s} \), but we can also write it as \( \frac{1}{s^{\frac{1}{2}}} \) or simplify the exponent fraction. Alternatively, we can present it as \( \frac{1}{s^{\frac{1}{2}}} \) which is \( \frac{1}{\sqrt{s}} \), but in the form with a positive exponent in the denominator, we can also write it as \( \frac{1}{s^{\frac{1}{2}}} \) or recognize that \( s^{\frac{1}{2}}=\sqrt{s} \), but the problem allows \( \frac{A}{B} \) form where \( A \) and \( B \) have no common variables. Since \( A = 1 \) and \( B = s^{\frac{1}{2}} \) (or \( \sqrt{s} \)), but we can also write the exponent as a reduced fraction. Wait, actually, from step 1, we had \( s^{-\frac{1}{2}}=\frac{1}{s^{\frac{1}{2}}} \), and \( \frac{1}{s^{\frac{1}{2}}} \) can be written as \( \frac{1}{s^{1/2}} \) or simplified the fraction \( -\frac{2}{4}=-\frac{1}{2} \), so the final form is \( \frac{1}{s^{\frac{1}{2}}} \) or \( \frac{1}{\sqrt{s}} \), but also, \( s^{\frac{1}{2}}=\sqrt{s} \), but the problem says "all exponents in your answer should be positive", so we can write it as \( \frac{1}{s^{\frac{1}{2}}} \) or \( \frac{1}{\sqrt{s}} \), but another way is to note that \( s^{-\frac{1}{2}}=\frac{1}{s^{\frac{1}{2}}} \), and \( \frac{1}{s^{\frac{1}{2}}} \) is equivalent to \( \frac{1}{s^{1/2}} \), which can be written as \( \frac{1}{s^{\frac{1}{2}}} \) or simplified the exponent: \( -\frac{2}{4}=-\frac{1}{2} \), so \( s^{-\frac{1}{2}}=\frac{1}{s^{\frac{1}{2}}} \).

Answer:

\( \frac{1}{s^{\frac{1}{2}}} \) (or \( \frac{1}{\sqrt{s}} \), but \( \frac{1}{s^{\frac{1}{2}}} \) is also correct as per the form requirements)