Question

simplify the following expression. write the result using positive exponents only.
$(-2x^{4}y^{-5})(3x^{-1}y^{2})$
$(-2x^{4}y^{-5})(3x^{-1}y^{2})=\square$ (type exponential notation with positive exponents.)

Explanation:

Step1: Multiply coefficients and like bases

Multiply the coefficients \(-2\) and \(3\), and use the rule \(a^m \cdot a^n = a^{m + n}\) for \(x\) and \(y\) terms.
For coefficients: \(-2\times3=-6\)
For \(x\) terms: \(x^{4}\cdot x^{-1}=x^{4 + (-1)} = x^{3}\)
For \(y\) terms: \(y^{-5}\cdot y^{2}=y^{-5 + 2}=y^{-3}\)
So we have \(-6x^{3}y^{-3}\)

Step2: Convert negative exponent to positive

Use the rule \(a^{-n}=\frac{1}{a^{n}}\), so \(y^{-3}=\frac{1}{y^{3}}\). Then rewrite the expression:
\(-6x^{3}\cdot\frac{1}{y^{3}}=\frac{-6x^{3}}{y^{3}}\) or \(-\frac{6x^{3}}{y^{3}}\)

Answer:

\(-\dfrac{6x^{3}}{y^{3}}\)