Question

simplify.
\\(\frac{a^{-8}b^{-3}c^{-5}}{a^{6}b^{9}c^{2}}\\)
write your answer using only positive exponents.

Explanation:

Step1: Use quotient rule for exponents

For the same base, when dividing, subtract the exponents: \( \frac{x^m}{x^n}=x^{m - n} \). Apply this to each variable:

For \( a \): \( a^{-8-6}=a^{-14} \)

For \( b \): \( b^{-3 - 9}=b^{-12} \)

For \( c \): \( c^{-5-2}=c^{-7} \)

So we have \( a^{-14}b^{-12}c^{-7} \)

Step2: Convert negative exponents to positive

Recall that \( x^{-n}=\frac{1}{x^n} \). So:

\( a^{-14}=\frac{1}{a^{14}} \), \( b^{-12}=\frac{1}{b^{12}} \), \( c^{-7}=\frac{1}{c^{7}} \)

Multiply these together: \( \frac{1}{a^{14}b^{12}c^{7}} \)

Answer:

\( \frac{1}{a^{14}b^{12}c^{7}} \)