Question

simplify. express your answer using positive exponents.
t^6 u^(-1) v^85 · t^(-1) u^(-5) v^(-1) · t^0 u^(-2) v^99

Explanation:

Step1: Combine like bases for \( t \)

Using the rule \( a^m \cdot a^n = a^{m + n} \), for the \( t \) terms: \( t^6 \cdot t^{-1} \cdot t^0 = t^{6 + (-1) + 0} = t^{5} \)

Step2: Combine like bases for \( u \)

For the \( u \) terms: \( u^{-1} \cdot u^{-5} \cdot u^{-2} = u^{-1 + (-5) + (-2)} = u^{-8} \). Using the rule \( a^{-n}=\frac{1}{a^n} \), this becomes \( \frac{1}{u^{8}} \)

Step3: Combine like bases for \( v \)

For the \( v \) terms: \( v^{85} \cdot v^{-1} \cdot v^{99} = v^{85 + (-1) + 99} = v^{183} \)

Step4: Multiply the results together

Multiply the simplified \( t \), \( u \), and \( v \) terms: \( t^{5} \cdot \frac{1}{u^{8}} \cdot v^{183} = \frac{t^{5}v^{183}}{u^{8}} \)

Answer:

\(\frac{t^{5}v^{183}}{u^{8}}\)