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Chapter 5: Problem 40

Use properties of exponents to simplify each expression. Express answers in exponential form with positive exponents only. Assume that any variables in denominators are not equal to zero. \(\left(-5 x^{4} y^{-3}\right)\left(4 x^{-1} y\right)\)

Short Answer

Expert verified
The simplified form of the expression is \(-20 x^{3} y^{-2}\).

Step by step solution

01

Multiply constants

First, multiply the constants \(-5\) and 4, which equals to \(-20\). This gives the expression \(-20 x^{4} y^{-3} x^{-1} y\).
02

Combine like terms

Next, combine \(x^{4}\) and \(x^{-1}\), using the property that \(x^{m} . x^{n} = x^{m+n}\), and similarly for \(y\). This results in \(-20 x^{4 - 1} y^{-3 + 1}\), which simplifies to\(-20 x^{3} y^{-2}\).
03

Change to positive exponents

The task requires the answer to have positive exponents only. Remember that \(x^{-n} = \frac{1}{x^n}\). Thus, follow this rule to change the expression to : \(-20 x^{3} (\frac{1}{y^{2}})\). Multiply \(-20\) by \(x^{3}\) and by \(\frac{1}{y^{2}}\), to get the final answer as \(-20 x^{3}y^{-2}\).

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