What if you knew that the volume of a sphere could be represented by the formula \begin{align*}V=\frac{4 \pi r^3}{3}\end{align*}, where \begin{align*}r\end{align*} is the radius, and that the radius of a sphere is 4 feet? How would the fraction bar in the formula affect the way that you found the sphere's volume? Which operation do you think the fraction bar represents? After completing this Concept, you'll be able to correctly interpret the fraction bar when finding the sphere's volume.

### Guidance

Fraction bars count as grouping symbols for PEMDAS, and should be treated as a set of parentheses. All numerators and all denominators can be treated as if they have invisible parentheses. When **real** parentheses are also present, remember that the innermost grouping symbols should be evaluated first. If, for example, parentheses appear in a numerator, they would take precedence over the fraction bar. If the parentheses appear outside of the fraction, then the fraction bar takes precedence.

#### Example A

Use the order of operations to simplify the following expression:

\begin{align*}\frac{z + 3}{4} - 1\end{align*} when \begin{align*}z = 2\end{align*}

**Solution:** Begin by substituting the appropriate value for the variable:

\begin{align*}\frac{(2+3)}{4} -1 = \frac{5}{4} -1\end{align*}.

Rewriting 1 as a fraction, the expression becomes:

\begin{align*}\frac{5}{4} - \frac{4}{4} = \frac{1}{4}\end{align*}

#### Example B

Use the order of operations to simplify the following expression:

\begin{align*}\left (\frac{a+2}{b+4} - 1 \right ) + b\end{align*} *when* \begin{align*}a = 3\end{align*} and \begin{align*}b = 1\end{align*}

**Solution:** Begin by substituting the appropriate value for the variable:

\begin{align*}\frac{(3+2)}{(1+4)} = \frac{5}{5} = 1\end{align*}

\begin{align*}(1 - 1) + b\end{align*} Substituting 1 for *b,* the expression becomes \begin{align*} 0 + 1 = 1\end{align*}

#### Example C

Use the order of operations to simplify the following expression:

\begin{align*}2 \times \left ( \frac{w + (x - 2z)}{(y + 2)^2} - 1 \right )\end{align*} *when* \begin{align*}w = 11, \ x = 3, \ y = 1\end{align*} and \begin{align*}z = -2\end{align*}

**Solution:** Begin by substituting the appropriate values for the variables:

\begin{align*}2 \left ( \frac{[11+(3-2(-2))]}{[(1+2)^2)]} - 1 \right ) = 2 \left ( \frac{(11+7)}{3^2} -1 \right ) = 2 \left (\frac{18}{9} - 1 \right )\end{align*}

Continue simplifying:

\begin{align*}2\left ( \frac{18}{9} - \frac{9}{9} \right ) = 2 \left ( \frac{9}{9} \right ) = 2(1)= 2\end{align*}

### Guided Practice

Use the order of operations to evaluate the following expression when \begin{align*}x = 6\end{align*} and \begin{align*}y = 2\end{align*}:

\begin{align*}\left (\frac{x-1}{y-1} \right )^2 + \frac{2x}{y^2}\end{align*}

**Solution:**

Begin by substituting in 6 for \begin{align*}x\end{align*} and 1 for \begin{align*}y\end{align*}.

\begin{align*}\left (\frac{6-1}{2-1} \right )^2 + \frac{2(6)}{2^2}\end{align*}

First, we work with what is inside the parentheses. There, we have a fraction so we have to simplify the fraction first, simplifying the numerator and then the denominator before dividing. We can simply the other fraction at the same time.

\begin{align*}\left (\frac{5}{1} \right )^2 + \frac{12}{4}\end{align*}

Next we simplify the fractions, and finish with exponents and then addition:

\begin{align*} (5 )^2 + 3=25+3=28.\end{align*}

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Order of Operations (14:23)

Use the order of operations to simplify the following expressions.

- \begin{align*}\frac{2 \cdot (3 + (2 - 1))}{4 - (6 + 2)} - (3 - 5)\end{align*}
- \begin{align*}\frac{(2+3)^2}{3-8} - \frac{3\cdot(10-4)}{7-4}\end{align*}

Evaluate the following expressions involving variables.

- \begin{align*}\frac{jk}{j + k}\end{align*} when \begin{align*}j = 6\end{align*} and \begin{align*}k = 12\end{align*}.
- \begin{align*}\frac{4x}{9x^2 - 3x + 1}\end{align*} when \begin{align*}x = 2\end{align*}
- \begin{align*}\frac{z^2}{x + y} + \frac{x^2}{x - y}\end{align*} 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*}

The formula to find the volume of a square pyramid is \begin{align*}V=\frac{s^2 (h)}{3}\end{align*}. Evaluate the volume for the given values.

- \begin{align*}s=4\ inches,h=18\ inches\end{align*}
- \begin{align*}s=10\ feet,h=50\ feet\end{align*}
- \begin{align*}h=7\ meters,s=12\ meters\end{align*}
- \begin{align*}h=27\ feet,s=13\ feet\end{align*}
- \begin{align*}s=16\ cm,h=90\ cm\end{align*}