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Expression Evaluation with Powers and Grouping Symbols

Evaluate expressions with parentheses and exponents.

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Use the Order of Operations to Evaluate Powers

License: CC BY-NC 3.0

Shirley is a math tutor and is working with middle-schooler Marcus once a week. One day, Marcus arrived with a problem that not even Shirley knew how to answer.

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

How can Shirley and Marcus work out the correct answer to this problem? Are there any rules they need to follow?

In this concept, you will learn to evaluate variable expressions with powers by applying the order of operations.

PEMDAS

The order of operations is a necessary tool to use when evaluating expressions. Remember the order in which the mathematical operations are performed affect your answer. You can recall the order of operations using the shorthand form PEMDAS.

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

Let’s look at an example.

Evaluate the following variable expression.

\begin{align*}8h^2+[51 \div (4 \cdot 4.25)]-5^2 \div 5 \end{align*}when\begin{align*}h=4\end{align*}

First, substitute the value \begin{align*}h=4\end{align*}into the expression. 

\begin{align*}8(4)^2+[51 \div (4 \cdot 4.25)] - 5^2 \div 5\end{align*} Next, expand the power:\begin{align*}(4)^2=4 \times 4\end{align*}and write the new expression. 

\begin{align*}8(4 \times 4)+[51 \div (4 \cdot 4.25)] - (5 \times 5) \div 5\end{align*}

Remember\begin{align*}-5^2=-(5 \times 5)\end{align*}since the base is 5. However,\begin{align*}(-5)^2=(-5 \times -5)\end{align*}since the base is\begin{align*}-5\end{align*}.

Next, perform the operations inside the grouping symbols.

First, multiply: \begin{align*}(4 \cdot 4.25)=17\end{align*}and write the new expression.

\begin{align*}8(4 \times 4)+[51 \div 17] -(5 \times 5) \div 5 \end{align*}

Then, divide: \begin{align*}51 \div 17=3\end{align*}and write the new expression. 

\begin{align*}8(4 \times 4)+3 - (5 \times 5) \div 5\end{align*}

Next, multiply: \begin{align*}(4 \times 4)=16\end{align*}and write the new expression.

\begin{align*}8(16)+3-(5 \times 5) \div 5\end{align*}

Then, multiply: \begin{align*}(5 \times 5)=25\end{align*} and write the new expression. 

\begin{align*}8(16)+3- 25 \div 5\end{align*}

Then, multiply: \begin{align*}8(16)=128\end{align*} to clear the parenthesis. Write the new expression. 

\begin{align*}128+3-25 \div 5\end{align*}

Next, divide: \begin{align*}25 \div 5= 5\end{align*}and write the new expression.

\begin{align*}128+3-5\end{align*}

Next, add: \begin{align*}128+3=131\end{align*}and write the new expression.

\begin{align*}131-5\end{align*}

Then, subtract:

\begin{align*}131-5=126\end{align*}

The answer is 126.

Let’s look at one more example. This example will be a variable expression with two different variables. There will be a value given for each of the variables. You simply substitute the given values for each variable into the expression and evaluate the new numerical expression.

Evaluate the following variable expression:

\begin{align*}4x^3-(3y \div 9)+12\end{align*}when \begin{align*}x=3\end{align*}and \begin{align*}y=3\end{align*}

First, substitute \begin{align*}x=3\end{align*}and \begin{align*}y=9\end{align*} into the expression. Write the new expression.

\begin{align*}4(3)^3-(3 \cdot 9 \div 9)+12\end{align*}

Next, expand the power: \begin{align*}(3)^3=(3 \times 3 \times 3)\end{align*}. Write the new expression.

\begin{align*}4(3 \times 3 \times 3) -(3 \cdot 9 \div 9)+12\end{align*}

Then, perform the operations inside the parenthesis.

First, multiply: \begin{align*}3 \cdot 9=27\end{align*}and write the new expression.

\begin{align*}4(3 \times 3 \times 3) -(27 \div 9)+12\end{align*}

Then, divide: \begin{align*}27 \div 9 =3\end{align*}and write the new expression.

\begin{align*}4(3 \times 3 \times 3) -3+12\end{align*}

Next, multiply: \begin{align*}(3 \times 3 \times 3)=27\end{align*}and write the new expression.

\begin{align*}4(27)-3+12\end{align*}

Then, multiply: \begin{align*}4(27)=108\end{align*}to clear the parenthesis. Write the new expression.

\begin{align*}108-3+12\end{align*}

Next, subtract: \begin{align*}108-3=5\end{align*}and write the new expression.

\begin{align*}105+12\end{align*}

Then, add:

\begin{align*}105+12=117\end{align*}

The answer is 117.

When you have powers included in variable and numerical expressions, you must use the order of operations to evaluate the expressions. No matter how complicated the problem seems to be, applying the order of operations as defined by PEMDAS will guide you to a correct answer as you evaluate the expression.

Examples

Example 1

Earlier, you were given a problem about Shirley and Marcus and their brain-bending problem.

They need to figure out the correct answer to the following problem.

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

Shirley and Marcus must apply the order of operations to evaluate the problem correctly.

First, substitute \begin{align*}x=2\end{align*}and\begin{align*}y=-1\end{align*}into the expression.

\begin{align*}(-11)^2+7(-1)^2+3(2)-19\end{align*}

Next, expand the powers.

\begin{align*}(-11 \times -11) + 7(-1 \times -1 )+3(2)-19\end{align*}

Next, do the multiplication inside the parentheses.

\begin{align*}121+7(1)+3(2)-19\end{align*}

Next, do the multiplication to clear the parentheses. 

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

Then, in order from left to right, perform the addition and subtraction.

\begin{align*}121+7=128+6=134-19=115\end{align*}

The answer is 115.

Example 2

Evaluate the following variable expression:

\begin{align*}2^3+4y+12\end{align*} when\begin{align*}y=3\end{align*}.
First, substitute\begin{align*}y=3\end{align*}into the variable expression. 
\begin{align*}2^3+4(3)+12\end{align*}
Next, expand the power: \begin{align*}2^3=2 \times 2 \times2 \end{align*} and write the new expression.
\begin{align*}2 \times 2 \times2+4(3)+12\end{align*} 
Next, multiply: \begin{align*}4(3)=12\end{align*} to clear the parenthesis. Write the new expression.
\begin{align*}2 \times 2 \times2+12+12\end{align*}

Next, multiply:\begin{align*}2 \times 2 \times 2 =8\end{align*} and write the new expression.

\begin{align*}8+12+12\end{align*}

Then, add:

\begin{align*}8+12=20+12=32\end{align*}

The answer is 32.

Example 3

Evaluate the following variable expression.

\begin{align*}-5^3+7y-30 \end{align*} when\begin{align*}y=9\end{align*}.

First, substitute \begin{align*}y=9\end{align*}into the variable expression. 

\begin{align*}-5^3+7(9)-30\end{align*}

Next, expand the power: \begin{align*}5^3=5 \times 5 \times 5\end{align*}and write the new expression.

\begin{align*}-5 \times 5 \times 5 + 7(9)-30\end{align*}

Next, multiply: \begin{align*}7(9)=63\end{align*} to clear the parenthesis. Write the new expression. 

\begin{align*}-5 \times 5 \times 5+63-30\end{align*}

Next, multiply: \begin{align*}5 \times 5 \times 5 =125\end{align*}and write the new expression. 

\begin{align*}-125+63-30\end{align*}

Then, subtract: \begin{align*}-125+63=-62\end{align*}and write the new expression.
\begin{align*}-62-30\end{align*}

Then, add:

\begin{align*}-62-30=-92\end{align*}
The answer is -92 .

Example 4

Evaluate the following variable expression:

\begin{align*}6x+7y+3^2\end{align*} when \begin{align*}x=4\end{align*}and\begin{align*} y=6\end{align*}

First, substitute \begin{align*} x=4\end{align*}and\begin{align*}y=6\end{align*}into the variable expression.

\begin{align*}6(4)+7(6)+3^2\end{align*}

Next, expand the power:\begin{align*}3^2=3 \times 3\end{align*}and write the new expression.

\begin{align*}6(4)+7(6)+3 \times 3\end{align*}

Next, multiply: \begin{align*}6(4)=24\end{align*}to clear the parenthesis. Write the new expression.

\begin{align*}24+7(6)+3 \times 3\end{align*}

Next, multiply: \begin{align*}7(6)=42\end{align*}to clear the parenthesis. Write the new expression.

\begin{align*}24+42+3 \times 3\end{align*}

Next multiply: \begin{align*}3 \times 3=9\end{align*}and write the new expression.

\begin{align*}24+42+9\end{align*}

Then, add:

\begin{align*}24+42=66+9=75\end{align*}

The answer is 75.

Review

Evaluate each numerical expression. Remember to follow the order of operations.

1. \begin{align*}3^2+[(5 \times 2) -3 ] -8 \times 2\end{align*}

2. \begin{align*}5^2 + (3+5)-6^2+2\end{align*}

3. \begin{align*} 6^3+5^2+25\end{align*}

4. \begin{align*}16(12)^3\end{align*}

5. \begin{align*}8^2-(2(3^3) \div 2)+(16 \times 5)\end{align*}

Evaluate each variable 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*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.7. 

 

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Vocabulary

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression 32^4, 32 is the base, and 4 is the exponent.

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.

Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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