Q11. - Algebra 1

Question

Simplify the given expression. State the excluded values of the variables.

 $\frac{\mathbf{16}{\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{3}}}{\mathbf{8}\mathbf{x}\mathbf{y}}$

Step-by-Step Solution

Answer

The simplified expression is $\mathbf{2}\mathit{x}{\mathit{y}}^{\mathbf{2}}$.

The excluded values are $\mathit{x}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{=}\mathbf{0}$.

Step 1. Define a rational function.

A function of the form $y=\frac{p}{q}$ is called a rational function, where $p$ and $q$ are the polynomials and $q\ne 0$ .

Step 2. Define quotient of power rule.

The quotient of power rule is given by

$\frac{a^m}{a^n}=a^{m-n}$ 

Where, $a$ is any non-zero real number.

Step 3. Define the excluded values for the rational function.

Any value of a variable that results in a denominator of zero in a rational function is excluded from the domain of the function and these values are called the excluded values of the rational function.

Step 4. Simplify the expression 16 x 2 y 3 8 x y .

Use quotient of power rule to simplify the expression $\frac{16x^2{y}^3}{8xy}$.

$\begin{array}{l}\frac{16x^2{y}^3}{8xy}=\left(\frac{16}{8}\right)\left(\frac{x^2}{x}\right)\left(\frac{y^3}{y}\right)\\ =\left(2\right)\left(x^{2-1}\right)\left(y^{3-1}\right)\\ =\left(2\right)\left(x^1\right)\left(y^2\right)\\ =2x{y}^2\end{array}$

Step 5. State the excluded values of the variables for the expression 16 x 2 y 3 8 x y .

For excluded value,

Denominator $=0$

 $\begin{array}{l}8xy=0\\ xy=\frac{0}{8}\\ xy=0\\ x=0,y=0\end{array}$ 

Therefore,

The simplified expression is $2x{y}^2$.

The excluded values are $x=0,y=0$.