Question

simplify. express your answer using positive exponents. \\( t^{-1} u^{-1} v \cdot t^{-5} u v^{-8} \cdot t u^{9} v^{-1} \\)

Explanation:

Step1: Group like bases

We have the expression \( t^{-1}u^{-1}v \cdot t^{-5}u v^{-8} \cdot t u^{9}v^{-1} \). Group the terms with the same base:
For base \( t \): \( t^{-1} \cdot t^{-5} \cdot t^{1} \)
For base \( u \): \( u^{-1} \cdot u^{1} \cdot u^{9} \)
For base \( v \): \( v^{1} \cdot v^{-8} \cdot v^{-1} \)

Step2: Apply exponent rule \( a^m \cdot a^n = a^{m + n} \) to each base

• For \( t \): \( t^{-1 + (-5)+ 1}=t^{-5} \)
• For \( u \): \( u^{-1 + 1+ 9}=u^{9} \)
• For \( v \): \( v^{1+(-8)+(-1)}=v^{-8} \)

Step3: Rewrite with positive exponents

Recall that \( a^{-n}=\frac{1}{a^{n}} \), so \( t^{-5}=\frac{1}{t^{5}} \) and \( v^{-8}=\frac{1}{v^{8}} \). The expression becomes \( \frac{u^{9}}{t^{5}v^{8}} \)

Answer:

\(\dfrac{u^{9}}{t^{5}v^{8}}\)