Question

simplify. express your answer using positive exponents.\\(\frac{6a^{-1}b^{6}c^{7}}{a^{-5}b^{-1}c^{-7}}\\)

Explanation:

Step1: Apply quotient rule for exponents

For the coefficient, it remains 6. For the variable \(a\), use the rule \(\frac{a^m}{a^n}=a^{m - n}\), so \(a^{-1-(-5)} = a^{-1 + 5}\). For \(b\), \(\frac{b^6}{b^{-1}}=b^{6-(-1)}=b^{6 + 1}\). For \(c\), \(\frac{c^7}{c^{-7}}=c^{7-(-7)}=c^{7 + 7}\).
\[
\frac{6a^{-1}b^{6}c^{7}}{a^{-5}b^{-1}c^{-7}}=6\times a^{-1-(-5)}\times b^{6-(-1)}\times c^{7-(-7)}
\]

Step2: Simplify exponents

Simplify each exponent: \(a^{-1 + 5}=a^{4}\), \(b^{6 + 1}=b^{7}\), \(c^{7 + 7}=c^{14}\).
\[
6\times a^{4}\times b^{7}\times c^{14}
\]

Answer:

\(6a^{4}b^{7}c^{14}\)