### Expressions Involving Fraction Bars

Fraction bars count as grouping symbols for **PEMDAS**, so we evaluate them in the first step of solving an expression. All numerators and all denominators can be treated as if they have invisible parentheses around them. When real parentheses are also present, remember that the innermost grouping symbols come first. If, for example, parentheses appear on a numerator, they would take precedence over the fraction bar. If the parentheses appear outside of the fraction, then the fraction bar takes precedence.

#### Evaluating Expressions

Use the order of operations to evaluate the following expression:

We substitute the value for

Although this expression has no parentheses, the fraction bar is also a grouping symbol, it has the same effect as a set of parentheses. We can write in the “invisible parentheses” for clarity:

Using **PEMDAS**, we first evaluate the numerator, 2 + 3 = 5. Now we have:

We can convert

Then evaluate the expression:

#### Evaluating Multi-Variable Expressions

Use the order of operations to evaluate the expression

Substitute the given values for

This expression effectively has nested parentheses (remember the effect of the fraction bar). The innermost grouping symbol is provided by the fraction bar. Evaluate the numerator

#### Using a Graphing Calculator

*Use a graphing calculator to evaluate the expression 3x2−4y2+x4(x+y)12 for x=2, y=−1.*

Store the values of **[STO]** **[STO]** **[ALPHA] + [KEY]**.) Input the expression in the calculator. When an expression includes a fraction, be sure to use parentheses:

Press **[ENTER]** to obtain the answer,

### Example

#### Example 1

Use the order of operations to evaluate the expression

Substitute the values for

This complicated expression has several layers of nested parentheses. One method for ensuring that we start with the innermost parentheses is to use more than one type of parentheses. Working from the inside, we leave the innermost grouping symbols as parentheses

### Review

For 1-3, use the order of operations to evaluate the expressions.

2⋅(3+(2−1))4−(6+2)−(3−5) 4+7(3)9−4+12−3⋅22 (22+5)252−42÷(2+1)

For 4-9, evaluate the expressions by substituting for the variables.

jkj+k whenj=6 andk=12 x+y2y−x whenx=2 andy=3

4x9x2−3x+1 whenx=2 z2x+y+x2x−y when \begin{align*}x = 1, \ y = -2,\end{align*} and \begin{align*}z = 4\end{align*}- \begin{align*}\frac { 4xyz } { y^2 - x^2 }\end{align*} when \begin{align*}x = 3, \ y = 2,\end{align*} and \begin{align*}z = 5\end{align*}
- \begin{align*}\frac { x^2 - z^2 } { xz - 2x(z - x)}\end{align*} when \begin{align*}x = -1 \end{align*} and \begin{align*}z = 3\end{align*}

For 10-14, evaluate each expression using a graphing calculator.

- \begin{align*}x^2 + 2x - xy\end{align*} when \begin{align*}x = 250\end{align*} and \begin{align*}y = -120\end{align*}
- \begin{align*}(xy - y^4)^2\end{align*} when \begin{align*}x = 0.02\end{align*} and \begin{align*}y = -0.025\end{align*}
- \begin{align*}\frac { x + y - z } { xy + yz + xz }\end{align*} when \begin{align*}x = \frac { 1 } { 2 }, \ y = \frac{3}{2},\end{align*} and \begin{align*}z = -1 \end{align*}
- \begin{align*}\frac{(x + y)^2}{4x^2 - y^2}\end{align*} when \begin{align*}x = 3\end{align*} and \begin{align*}y = -5\end{align*}
- \begin{align*}\frac{(x - y)^3}{x^3 - y} + \frac{(x + y)^2}{x + y^4}\end{align*} when \begin{align*}x = 4\end{align*} and \begin{align*}y = -2\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 1.5.