Do you know how to evaluate a variable expression when it includes powers? Take a look at this problem.

\begin{align*}(-11)^2 + 7y^2 + 3x - 19\end{align*} for \begin{align*}x=2,y=-1\end{align*}

Evaluating dilemmas like this one is just what this Concept is all about. Pay attention and you will be able to work through this problem at the end of the Concept.

### Guidance

Did you know that you can apply the order of operations to expressions that have powers in them?

Let's look at how to do this.

**To do this, we are going to need to refer back to the order of operations.**

**Order of Operations**

**P parentheses or grouping symbols**

**E exponents**

**MD multiplication and division in order from left to right**

**AS addition and subtraction in order from left to right**

**Now look at the E. That E refers to exponents and powers and evaluating exponents in the order of operations. You can see that you evaluate the powers right after the grouping symbols.**

It is a bit like working on a puzzle. Here is an expression that needs evaluating.

Evaluate the expression \begin{align*}8h^2 + [51 \div (4 \cdot 4.25)] - 52 \div 5\end{align*}. Let \begin{align*}h=4\end{align*}.

It does look complicated, but if it helps, think of this as a series of steps. The order of operations is your guide. If you follow the order of operations then working through a problem such as this one becomes much easier.

\begin{align*}& P && Step \ 1: \ \text{Substitute} \ 4 \ \text{for} \ ''h.''\\ & && 8h^2 + [51 \div (4 \cdot 4.25)] - 52 \div 5\\ & && 8(4)^2 + [51 \div (4 \cdot 4.25)] - 52 \div 5\\ & && Step \ 2: \ \text{Remember PEMDAS. Therefore, perform the operation inside the}\\ & && \text{grouping symbols first. Recall that order of operations must be followed}\\ & && \text{inside grouping symbols also. In this case, multiply} \ 4 \times 4.25 \ \text{before dividing}\\ & && \text{by} \ 51.\\ & && 8(4)^2 + [51 \div (4 \cdot 4.25)] - 52 \div 5\\ & && 8(4)^2 + [51 \div 17] - 52 \div 5\\ & && 8(4)^2 + 3 - 52 \div 5\\ & E && Step \ 3: \ \text{The next step in order of operations is to simplify the numbers with}\\ & && \text{exponents.}\\ & && 8(4)^2+ 3 - 52 \div 5\\ & && 8(4 \cdot 4) + 3 - 5 \cdot 5 \div 5\\ & && 8(16) + 3 - 25 \div 5\\ & M && Step \ 4: \ \text{Multiply}\\ & && 8(16) + 3 - 25 \div 5\\ & && 128 + 3 - 25 \div 5\\ & D && Step \ 5: \ \text{Divide}\\ & && 128 + 3 - 25 \div 5\\ & && 128 + 3 - 5\\ & A && Step \ 6: \ \text{Add}\\ & && 128 + 3 - 5\\ & && 131 - 5\\ & S && Step \ 6: \ \text{Subtract}\\ & && 131 - 5 = 126\end{align*}

**The answer is 126.**

**We can also evaluate variable expressions that have more than one variable. Notice that a different value has been given for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. You simply substitute the given values into each expression and evaluate it for the quantity of the expression.**

Evaluate the expression \begin{align*}4x^3 - (3y \div 9) + 12\end{align*}. Let \begin{align*}x=3\end{align*} and \begin{align*}y=9\end{align*}.

\begin{align*}& 4x^3 - (3y \div 9) + 12 \ (\text{Substitute the variables})\\ & 4(3)^3 - [(3 \times 9) \div 9] + 12 \ (\text{Parentheses})\\ & 4(3)^3 - [27 \div 9] + 12\\ & 4(3)^3 - 3 + 12 \ (\text{Exponents})\\ & 4(3 \times 3 \times 3) - 3 + 12\\ & 4(27) - 3 + 12 \ (\text{Multiply})\\ & 108 - 3 + 12 \ (\text{Add and then Subtract from left to right})\\ & 105 + 12\\ & 117\end{align*}

**The answer is 117.**

**When you have variable and numerical expressions with powers in them, you can use the order of operations to evaluate the expressions. Remember not to get stuck if the problem seems complicated. Stick to the order of operations and you will be able to evaluate the expression.**

#### Example A

Evaluate the expression \begin{align*}2^3 + 4y + 12\end{align*} for \begin{align*}y=3\end{align*}.

**Solution: \begin{align*}32\end{align*}**

#### Example B

Evaluate the expression \begin{align*}-5^3 + 7y - 30\end{align*} for \begin{align*}y=9\end{align*}.

**Solution: \begin{align*}-92\end{align*}**

#### Example C

Evaluate the expression \begin{align*}6x + 7y + 3^2\end{align*} for \begin{align*}x=4,y=6\end{align*}.

**Solution: \begin{align*}75\end{align*}**

Now let's go back to the dilemma from the beginning of the Concept. Evaluate this expression.

\begin{align*}(-11)^2 + 7y^2 + 3x - 19\end{align*} for \begin{align*}x=2,y=-1\end{align*}

First, let's substitute the given values into the expression for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

\begin{align*}(-11)^2 + 7(-1)^2 + 3(2) - 19\end{align*} for \begin{align*}x=2,y=-1\end{align*}

Now we can evaluate the powers. Here is our answer so far.

\begin{align*}121 + 7 + 6 - 19\end{align*}

\begin{align*}115\end{align*}

**This is our final answer.**

### Vocabulary

- Numerical Expression
- a group of numbers and operations used to represent a quantity without an equals sign.

- Variable Expression
- a group of numbers, operations and variables used to represent a quantity without an equals sign.

- Powers
- the value of a base and an exponent.

- Base
- the regular sized number that the exponent works upon.

- Exponent
- the little number that tells you how many times to multiply the base by itself.

### Guided Practice

Here is one for you to try on your own.

Evaluate the expression \begin{align*}-12^3 + 7y^2 + 12\end{align*} for \begin{align*}y=6\end{align*}.

**Solution**

*Step 1:* Before performing the order of operations, substitute 6 for “\begin{align*}y\end{align*}.”

\begin{align*}-12^3 + 7(6)^2 + 12\end{align*}

*Step 2:* Perform the calculations inside the parentheses.

\begin{align*}-12^3 + 7(36) + 12\end{align*}

*Step 3:* Perform the calculations with exponents.

\begin{align*}& -12^3 + 7(36) + 12\\ & -1,728 + 7(36) + 12\end{align*}

*Step 4:* Multiply

\begin{align*}& -1,728 + 7(36) + 12\\ & -1,728 + 252 + 12\end{align*}

*Step 5:* Add

\begin{align*}-1,728 + 252 + 12 = - 1,464\end{align*}

**The answer is -1,464.**

### Video Review

Khan Academy Introduction to the Order of Operations

### Practice

Directions: Evaluate each expression. Remember to follow the order of operations.

- \begin{align*}3^2 + [(5 \cdot 2) - 3] - 8 \cdot 2\end{align*}
- \begin{align*}5^2 + (3 + 5) - 6^2 + 2\end{align*}
- \begin{align*}6^3 + 5^2 + 25\end{align*}
- \begin{align*}16(12^3)\end{align*}
- \begin{align*}8^2 - (2(3^3) \div 2) + (16 \cdot 5)\end{align*}

Directions: Evaluate each expression by substituting the given value into each expression. Remember to follow the order of operations.

6. \begin{align*}-2^3 + 7y + 1\end{align*} for \begin{align*}y=6\end{align*}.

7. \begin{align*}-12 + 7x^2 - 8\end{align*} for \begin{align*}x=6\end{align*}.

8. \begin{align*}14 + 7y^2 + 22\end{align*} for \begin{align*}y=3\end{align*}.

9. \begin{align*}18x + 7y + 12\end{align*} for \begin{align*}x=3,y=6\end{align*}.

10. \begin{align*}-6^3 + 7x^2 - 18\end{align*} for \begin{align*}x=5\end{align*}.

11. \begin{align*}45 + 8y + 3^3\end{align*} for \begin{align*}y=5\end{align*}.

12. \begin{align*}-3^3 + 8x -2^2\end{align*} for \begin{align*}x=7\end{align*}.

13. \begin{align*}(-12)^2 + 7y -4^2\end{align*} for \begin{align*}y=6\end{align*}.

14. \begin{align*}-4^3 + 9x + 11\end{align*} for \begin{align*}x=4\end{align*}.

15. \begin{align*}(-7)^2 + 7x^2 + 12^2\end{align*} for \begin{align*}y=2\end{align*}.

16. \begin{align*}-45 + 7^2 - x^3\end{align*} for \begin{align*}x=4\end{align*}.