Question

question 1-8
multiply. express your solution with positive exponents.
$(-2a^{-5}b^6)(7a^4b^{-3})$
\\(\frac{-14b^3}{a}\\)
\\(\frac{-14}{ab^3}\\)
\\(\frac{5b^5}{a}\\)
\\(\frac{5}{ab^3}\\)

Explanation:

Step1: Multiply coefficients and like bases

Multiply the coefficients \(-2\) and \(7\), and use the product rule for exponents \(x^m \cdot x^n = x^{m + n}\) for \(a\) and \(b\) terms.
\((-2a^{-5}b^{6})(7a^{4}b^{-3}) = (-2\times7) \cdot (a^{-5} \cdot a^{4}) \cdot (b^{6} \cdot b^{-3})\)

Step2: Calculate coefficients and exponents

Calculate the coefficient: \(-2\times7=-14\)
For \(a\) terms: \(a^{-5 + 4}=a^{-1}\)
For \(b\) terms: \(b^{6+(-3)} = b^{3}\)
So we have \(-14a^{-1}b^{3}\)

Step3: Convert negative exponent to positive

Use the rule \(x^{-n}=\frac{1}{x^{n}}\) to convert \(a^{-1}\) to \(\frac{1}{a}\)
\(-14a^{-1}b^{3}=\frac{-14b^{3}}{a}\)

Answer:

\(\boldsymbol{\frac{-14b^{3}}{a}}\) (corresponding to the first option: \(\frac{-14b^{3}}{a}\))