Chapter A, Problem 1E

To determine

To calculate: The simplified form of the expression $\left(a^4{b}^{-3}\right)\left(a^{-6}{b}^2\right)$.

Answer to Problem 1E

Solution:

The simplified expression of $\left(a^4{b}^{-3}\right)\left(a^{-6}{b}^2\right)$ is $\frac{1}{a^{2}b}$.

Explanation of Solution

Given Information:

The expression is $\left(a^4{b}^{-3}\right)\left(a^{-6}{b}^2\right)$.

Formula used:

The definition of negative exponents states that for any nonzero real number $a$ and integer n,

$a^{-n}=\frac{1}{a^n}$.

The Product rule states that for any real number a, and integers m and n,

$a^m\cdot a^n=a^{m+n}$.

Calculation:

Consider the expression.

$\left(a^4{b}^{-3}\right)\left(a^{-6}{b}^2\right)$

Apply the product rule to simplify the expression.

$\begin{array}{c}\left(a^4{b}^{-3}\right)\left(a^{-6}{b}^2\right)=\left(a^4\cdot a^{-6}\right)\left(b^{-3}\cdot b^2\right)\\ =\left(a^{4+\left(-6\right)}\right)\left(b^{\left(-3\right)+2}\right)\end{array}$

Add the exponents.

$\begin{array}{c}\left(a^4{b}^{-3}\right)\left(a^{-6}{b}^2\right)=\left(a^{-2}\right)\left(b^{-1}\right)\\ =a^{-2}{b}^{-1}\end{array}$

Use the definition of the negative exponents to solve further.

$\begin{array}{c}\left(a^4{b}^{-3}\right)\left(a^{-6}{b}^2\right)=\left(\frac{1}{a^2}\right)\left(\frac{1}{b^1}\right)\\ =\frac{1}{a^{2}b}\end{array}$

Therefore, the simplified expression of $\left(a^4{b}^{-3}\right)\left(a^{-6}{b}^2\right)$ is $\frac{1}{a^{2}b}$.