Chapter 7, Problem 1RP

In all exercises assume that the variables represent positive real numbers unless otherwise stated. Simplify using only positive exponents in your

answers.

$\left(3x{y}^{1/2}\right)\left(5x^2{y}^{-3}\right)$

To determine

To calculate: Simplified value the following expression $\left(3x{y}^{1/2}\right)\left(5x^2{y}^{-3}\right)$ using only positive exponents.

Answer to Problem 1RP

Solution:

The simplified expression of $\left(3x{y}^{1/2}\right)\left(5x^2{y}^{-3}\right)$ is $\frac{15x^3}{y^{5/2}}$.

Explanation of Solution

Formula used:

The rules of exponents used are:

$\begin{array}{c}{x}^m\cdot x^n=x^{m+n}\\ x^{-n}=\frac{1}{x^n}\end{array}$

Calculation:

Consider the expression $\left(3x{y}^{1/2}\right)\left(5x^2{y}^{-3}\right)$.

Apply the product rule of exponent to simplify the expression as,

$\begin{array}{c}\left(3x{y}^{1/2}\right)\left(5x^2{y}^{-3}\right)=\left(3\cdot 5\right)\left(x^{1+2}\right)\left(y^{1/2-3}\right)\\ =15x^3{y}^{-5/2}\end{array}$

Use the rule of negative exponent for variable y to simplify into only positive exponents as,

$\begin{array}{c}\left(3x{y}^{1/2}\right)\left(5x^2{y}^{-3}\right)=15x^3{y}^{-5/2}\\ =\frac{15x^3}{y^{5/2}}\end{array}$

Thus, the simplified expression is $\frac{15x^3}{y^{5/2}}$.