Question

simplify. express your answer using positive exponents.\\(\frac{s^{-2}t^{-1}}{s^{-3}t^{-4}\cdot s^{-5}t^{-1}}\\)

Explanation:

Step1: Simplify denominator exponents

For the denominator, use the rule \(a^m \cdot a^n = a^{m + n}\). For \(s\): \(-3 + (-5)=-8\), for \(t\): \(-4 + (-1)=-5\). So denominator is \(s^{-8}t^{-5}\).

Step2: Divide numerator and denominator

Use the rule \(\frac{a^m}{a^n}=a^{m - n}\). For \(s\): \(-2-(-8)=6\), for \(t\): \(-1-(-5)=4\). So we get \(s^{6}t^{4}\) in reciprocal? Wait, no: \(\frac{s^{-2}t^{-1}}{s^{-8}t^{-5}}=s^{-2 - (-8)}t^{-1 - (-5)}=s^{6}t^{4}\)? Wait, no, wait: numerator is \(s^{-2}t^{-1}\), denominator after step1 is \(s^{-8}t^{-5}\). So \(\frac{s^{-2}}{s^{-8}} = s^{-2+8}=s^{6}\), \(\frac{t^{-1}}{t^{-5}}=t^{-1 +5}=t^{4}\). Wait, no, subtraction: \(m - n\) when dividing. So \(-2 - (-8)=6\), \(-1 - (-5)=4\). So the result is \(s^{6}t^{4}\)? Wait, no, wait the original denominator is \(s^{-3}t^{-4} \cdot s^{-5}t^{-1}\), so adding exponents: \(s^{-3-5}=s^{-8}\), \(t^{-4-1}=t^{-5}\). Then numerator is \(s^{-2}t^{-1}\), so dividing: \(\frac{s^{-2}t^{-1}}{s^{-8}t^{-5}} = s^{-2 - (-8)} t^{-1 - (-5)} = s^{6} t^{4}\)? Wait, no, that can't be. Wait, no, the rule is \(\frac{a^m}{a^n}=a^{m - n}\), so for \(s\): \(-2 - (-8)=6\), for \(t\): \(-1 - (-5)=4\). So the expression simplifies to \(s^{6}t^{4}\)? Wait, no, wait I think I messed up. Wait, numerator: \(s^{-2}t^{-1}\), denominator: \(s^{-3}t^{-4} \cdot s^{-5}t^{-1}=s^{-8}t^{-5}\). Then \(\frac{s^{-2}t^{-1}}{s^{-8}t^{-5}} = s^{-2 + 8} t^{-1 + 5}\) (because dividing with exponents is subtracting, but subtracting a negative is adding). So \(s^{6}t^{4}\). Wait, but let's check again. Let's write all steps:

Original expression: \(\frac{s^{-2}t^{-1}}{s^{-3}t^{-4} \cdot s^{-5}t^{-1}}\)

First, simplify the denominator by adding exponents of like bases:

For \(s\) in denominator: \(-3 + (-5) = -8\), so \(s^{-8}\)

For \(t\) in denominator: \(-4 + (-1) = -5\), so \(t^{-5}\)

So denominator is \(s^{-8}t^{-5}\)

Now, divide numerator (\(s^{-2}t^{-1}\)) by denominator (\(s^{-8}t^{-5}\)):

Using \(\frac{a^m}{a^n} = a^{m - n}\):

For \(s\): \(-2 - (-8) = -2 + 8 = 6\), so \(s^{6}\)

For \(t\): \(-1 - (-5) = -1 + 5 = 4\), so \(t^{4}\)

So the simplified expression is \(s^{6}t^{4}\)? Wait, no, wait, no—wait, the numerator is \(s^{-2}t^{-1}\), denominator is \(s^{-8}t^{-5}\), so when we divide, it's \(s^{-2 - (-8)} t^{-1 - (-5)} = s^{6} t^{4}\). Wait, but let's verify with positive exponents from the start.

Alternative approach: Move all negative exponents to the other side. \(s^{-2}=\frac{1}{s^2}\), \(t^{-1}=\frac{1}{t}\), \(s^{-3}=\frac{1}{s^3}\), \(t^{-4}=\frac{1}{t^4}\), \(s^{-5}=\frac{1}{s^5}\), \(t^{-1}=\frac{1}{t}\).

So numerator: \(\frac{1}{s^2 t}\)

Denominator: \(\frac{1}{s^3 t^4} \cdot \frac{1}{s^5 t} = \frac{1}{s^8 t^5}\)

Now, dividing by a fraction is multiplying by its reciprocal: \(\frac{1}{s^2 t} \cdot s^8 t^5 = s^{8 - 2} t^{5 - 1} = s^6 t^4\). Yes, that's correct.

Answer:

\(s^{6}t^{4}\)