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# 1.6: Evaluate Numerical and Variable Expressions Involving Powers

Difficulty Level: Basic Created by: CK-12
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Practice Algebra Expressions with Exponents

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Casey missed three days of school this week because of a nasty cold. When he returned to school on Thursday he asked his math teacher what he had missed. Miss Brown wrote a problem on a sheet of paper and handed it to Casey. When he arrived home from school Casey looked at his homework which was the following problem:

54+(2)4+12\begin{align*}{5}^{4}+{(-2)}^{4}+12\end{align*}

Casey didn’t know what to do with 54\begin{align*}{5}^{4}\end{align*}and (2)4\begin{align*}{(-2)}^{4}\end{align*}. How can he rewrite the problem so that he understands what operations to perform?

In this concept, you will learn how to evaluate numerical expressions involving powers.

### Evaluating Expressions Involving Powers

A numerical expression is an expression made up only of numbers and operations but an expression written with a variable in it, is called a variable expression.

Both a numerical expression and a variable expression can include powers.

A power is the result of multiplying a number by itself one or more times. The number 16 is the fourth power of 2.

2×2×2×2=16\begin{align*}2\times 2\times 2\times 2=16\end{align*}. The ‘fourth power of 2’ or ‘2 to the power of 4’ can be written as 24\begin{align*}{2}^{4}\end{align*}. The number raised to the power is called the base and the number expressing the power is called the exponent. The exponent tells you how many times to multiply the base times itself.63=6×6×6\begin{align*}{6}^{3}=6 \times 6 \times 6\end{align*}.

Let’s look at an example.

Evaluate the following expression:

63+52+25\begin{align*}{6}^{3}+{5}^{2}+25\end{align*}

First, expand the powers to see the operations you need to perform.

63=6×6×6 and 52=5×5\begin{align*}{6}^{3}=6 \times 6 \times 6 \ \text{and} \ {5}^{2}= 5 \times 5\end{align*}Next, write the expression in expanded form.

6×6×6+5×5+25\begin{align*}6 \times 6 \times 6 + 5 \times 5+ 25\end{align*}

Next, apply the order to operations (PEMDAS) to evaluate the expanded expression.

Then, in order from left to right, multiply: 6×6=36×6=216\begin{align*}6 \times 6 = 36 \times 6 = 216\end{align*}and write the new expression.

216+5×5+25\begin{align*}216 + 5 \times 5 + 25\end{align*}

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

216+25+25\begin{align*}216+25+25\end{align*}

Next, from left to right, add: 216+25=241\begin{align*}216+25=241\end{align*}and write the new expression.

241+25\begin{align*}241+25\end{align*}

Then, add:241+25=266\begin{align*}241+25=266\end{align*}

Let’s look at another example.

62+15+3311\begin{align*}{6}^{2}+15+{3}^{3}-11\end{align*}

First, expand the powers to see the operations you need to perform.

62=6×6 and 33=3×3×3\begin{align*}{6}^{2}=6 \times 6 \ \text{and} \ {3}^{3}=3 \times 3 \times 3 \end{align*}

Next, write the expression in expanded form.

6×6+15+3×3×311\begin{align*}6 \times 6 + 15+ 3 \times 3 \times 3 -11\end{align*}

Next, apply the order to operations (PEMDAS) to evaluate the expanded expression.

Then, in order from left to right multiply: 6×6=36\begin{align*}6 \times 6 =36\end{align*}and write the new expression.

36+15+3×3×311\begin{align*}36+15+3 \times 3 \times 3 -11\end{align*}

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

36+15+2711\begin{align*}36+15+27-11\end{align*}

Next, from left to right, add: 36+15=51\begin{align*}36+15=51\end{align*}and write the new expression.

51+2711\begin{align*}51+27-11\end{align*}

Then, add:51+2778\begin{align*}51+27-78\end{align*}and write the new expression.

7811\begin{align*}78-11\end{align*}

Next, subtract:7811=67\begin{align*}78-11=67\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Casey and his confusing homework.

Casey needs to expand the powers so he can see what operations he has to do to evaluate the expression.

First, evaluate the two powers given in the problem.

54=5×5×5×5=625 and (2)4=2×2×2×2=16\begin{align*}5^4= 5 \times 5 \times 5 \times 5 =625 \ \text{and} \ (-2)^4=-2 \times -2 \times-2 \times -2 =16 \end{align*}

Next, replace the powers in the expression with their values.

625+16+12\begin{align*}625+16+12\end{align*}Then, add.

625+16=641+12=653\begin{align*}625+16=641+12=653\end{align*}

#### Example 2

Evaluate the following variable expression whenx=4\begin{align*}x=4\end{align*}

2x312\begin{align*}2x^3-12\end{align*}

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

2(4)312\begin{align*}2(4)^3-12\end{align*}

Next, expand the power.

(4)3=(4×4×4)\begin{align*}(4)^3=(4 \times 4 \times 4)\end{align*}Next, write the expression in expanded form.

2(4×4×4)12\begin{align*}2(4 \times 4 \times 4 ) -12\end{align*}

Then, perform the operation in the parenthesis.

Multiply:4×4=16×4=64\begin{align*}4 \times 4 = 16 \times 4 =64\end{align*}and write the new expression.

2(64)12\begin{align*}2(64)-12\end{align*}

Next, multiply: 2(64)=128\begin{align*}2(64)=128\end{align*}and write the new expression.

12812\begin{align*}128-12\end{align*}

Then, subtract.

12812=116\begin{align*}128-12=116\end{align*}

#### Example 3

Evaluate the following numerical expression.

2024+1+33\begin{align*}20-2^4+1+3^3\end{align*}

First, expand the powers.

2×2×2×2 and 3×3×3\begin{align*}2 \times 2 \times 2 \times 2 \ \text{and} \ 3 \times 3 \times 3\end{align*}

Next, write the expression in expanded form.

202×2×2×2+1+3×3×3\begin{align*}20- 2 \times 2 \times 2 \times 2 + 1+ 3 \times 3 \times 3\end{align*}

Then, in order from left to right multiply: 2×2×2×2=16\begin{align*}2 \times 2 \times 2 \times 2=16\end{align*}and write the new expression.

2016+1+3×3×3\begin{align*}20-16+1+ 3 \times 3 \times 3\end{align*}

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

2016+1+27\begin{align*}20-16+1+27\end{align*}

Then, in order from left to write, subtract: 2016=4\begin{align*}20-16=4\end{align*} and write the new expression.

4+1+27\begin{align*}4+1+27\end{align*}

Next, add: 4+1=5\begin{align*}4+1=5\end{align*}and write the new expression.

5+27\begin{align*}5+27\end{align*}

Then add: 5+27=32\begin{align*}5+27=32\end{align*}

#### Example 4

Evaluate the following variable expression when m=3\begin{align*}m=3\end{align*}andn=2\begin{align*}n=2\end{align*}

n5+3m215\begin{align*}n^5+3m^2-15\end{align*}

First, substitute m=3\begin{align*}m=3\end{align*}andn=2\begin{align*}n=2\end{align*}into the variable expression.

25+3(3)215\begin{align*}2^5+3(3)^2-15\end{align*}Next, expand the powers:25=(2×2×2×2×2) and (3)2=(3×3)\begin{align*}2^5=( 2 \times 2 \times 2 \times 2 \times 2) \ \text{and} \ (3)^2 =(3 \times 3)\end{align*}

Then, write the expression in expanded form.

2×2×2×2×2+3(3×3)15\begin{align*}2 \times 2 \times 2 \times 2 \times 2 +3 (3 \times 3)-15\end{align*}

Then, perform the operation in the parenthesis.

First, multiply: (3×3)=(9)\begin{align*}(3 \times 3)=(9)\end{align*}and write the new expression.

2×2×2×2×2+3(9)15\begin{align*}2 \times 2 \times 2 \times 2 \times 2 + 3(9)-15\end{align*}

Next, multiply: 3(9)=27\begin{align*}3(9)=27\end{align*}to clear the parenthesis. Write the new expression.

2×2×2×2×2+2715\begin{align*}2 \times 2 \times 2 \times 2 \times 2 +27-15\end{align*}

Next, multiply:  2×2=4×2=8×2=16×2=32\begin{align*}2 \times 2 = 4 \times 2 = 8 \times 2 =16 \times 2 =32 \end{align*}Write the new expression.

32+2715\begin{align*}32+27-15\end{align*}

Next, add: 32+27=59\begin{align*}32+27=59\end{align*}and write the new expression.

5915\begin{align*}59-15\end{align*}Then, subtract: 5915=44\begin{align*}59-15=44\end{align*}

### Review

Expand and evaluate each power.

1. 33\begin{align*}3^3\end{align*}

2. 42\begin{align*}4^2\end{align*}

3. (2)4\begin{align*}(-2)^4\end{align*}

4. (8)2\begin{align*}(-8)^2\end{align*}

5. 53\begin{align*}5^3\end{align*}

6. 26\begin{align*}2^6\end{align*}

7. (9)2\begin{align*}(-9)^2\end{align*}

8. (2)6\begin{align*}(-2)^6\end{align*}

Evaluate each numerical expression. Remember to apply PEMDAS to evaluate the expression accurately.

9. 62+22\begin{align*}6^2+22\end{align*}

10. (3)3+18\begin{align*}(-3)^3+18\end{align*}

11. 23+164\begin{align*}2^3+16-4\end{align*}

12. (5)219\begin{align*}(-5)^2-19\end{align*}

13. (7)2+522\begin{align*}(-7)^2+52-2\end{align*}

14. 18+923\begin{align*}18+9^2-3\end{align*}

15. 2233+7\begin{align*}22-3^3+7\end{align*}

Evaluate each variable expression using the given values.

16. 6a+422\begin{align*}6a+4^2-2\end{align*}, when a=3\begin{align*}a=3\end{align*}

17. a3+14\begin{align*}a^3+14\end{align*}, when a=6\begin{align*}a=6\end{align*}

18. 2a216\begin{align*}2a^2-16\end{align*}, when a=4\begin{align*}a=4\end{align*}

19. 5b3+12\begin{align*}5b^3+12\end{align*}, when b=2\begin{align*}b=-2\end{align*}

20. 2x2+52\begin{align*}2x^2+52\end{align*}, when x=4\begin{align*}x=4\end{align*}

To see the Review answers, open this PDF file and look for section 1.6.

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### Vocabulary Language: English

TermDefinition
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.
Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Integer The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Numerical expression A numerical expression is a group of numbers and operations used to represent a quantity.
Parentheses Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
Power The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".
substitute In algebra, to substitute means to replace a variable or term with a specific value.
Variable Expression A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.
Volume Volume is the amount of space inside the bounds of a three-dimensional object.

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