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\(x\)
\(n\)

Evaluate Expressions
With Exponents

Master the power of repeated multiplication and learn to read, write, and evaluate expressions with whole-number exponents.

6th Grade Math

What Is an Exponent?

An exponent tells you how many times to multiply a number by itself. Instead of writing the same number over and over, we use a small raised number as a shortcut.

\(2\)
\(5\)
\(2 \times 2 \times 2 \times 2 \times 2\)
\(= 32\)
 Base Exponent

How to Read Exponents

There are special names for the most common exponents, and a general pattern for all others:

\( n^2 \)
"n squared"
\( n^3 \)
"n cubed"
\( n^4 \)
"n to the fourth power"
\( n^k \)
"n to the k-th power"

Why Do Exponents Matter?

Exponents show up everywhere in math and science. They let us express very large (or very small) numbers in a compact way.

Area
A square with side \( s \) has area \( s^2 \).
Volume
A cube with edge \( e \) has volume \( e^3 \).
Growth
Bacteria doubling every hour: \( 2^n \) after \( n \) hours.

The Core Rule of Exponents

Definition For any base b and whole-number exponent n:

$$ b^n = \underbrace{b \times b \times b \times \cdots \times b}_{n \text{ factors}} $$

Special Cases: \( b^1 \) and \( b^0 \)

Exponent = 1 Only 1 factor of b Exponent = 0 Zero factors = 1
\(7^1 = 7\)
\(100^1 = 100\)
\(1^1 = 1\)
\(5^0 = 1\)
\(99^0 = 1\)
\(1000^0 = 1\)

Exponents vs Multiplication

Students often confuse \( 3^4 \) with \( 3 \times 4 \). They look similar but mean very different things:

Exponent: \( 3^4 \) Multiply: \( 3 \times 4 \)
Meaning \( 3 \times 3 \times 3 \times 3 \) \( 3 + 3 + 3 + 3 \)
Result 81 12
Growth Repeated multiplication Repeated addition

How to Evaluate Step by Step

1
Identify the base and the exponent in the expression.
2
Expand the expression: write the base as a factor the number of times shown by the exponent.
3
Multiply from left to right, one pair at a time.
4
Write the final product as your answer.

Example 1: Evaluate \( 4^3 \)

STEP 1: Identify Base = 4, Exponent = 3 STEP 2: Expand
\(4 \times 4 \times 4\)
 STEP 3: Multiply
\(4 \times 4 = 16\), then \(16 \times 4 = 64\)
\(= 64\)

Example 2: Evaluate \( 2^6 \)

Doubling Chain for \(2^6\)
\(2\)
\(2^1\)
 ×2
\(4\)
\(2^2\)
 ×2
\(8\)
\(2^3\)
 ×2
\(16\)
\(2^4\)
 ×2
\(32\)
\(2^5\)
 ×2
\(64\)
\(2^6\)
\(2 \times 2 \times 2 \times 2 \times 2 \times 2\)
\(= 4 \times 4 \times 4 = 64\)
 Tip: group pairs to simplify!

Example 3: Evaluate \( 3 + 5^2 \times 2 \)

When an expression has exponents mixed with other operations, always follow the order of operations (PEMDAS): exponents come before multiplication, and multiplication before addition.

1
Exponents first: \( 5^2 = 25 \)
2
Multiply: \( 25 \times 2 = 50 \)
3
Add: \( 3 + 50 = 53 \)

Example 4: \( (2 + 3)^3 \)

When a sum is inside parentheses and raised to a power, always simplify inside the parentheses first, then apply the exponent.

1
Parentheses first: \( 2 + 3 = 5 \)
2
Apply exponent: \( 5^3 = 5 \times 5 \times 5 \)
3
Multiply: \( 5 \times 5 = 25 \), then \( 25 \times 5 = 125 \)

Powers of 10

Powers of 10 are especially useful because the exponent tells you exactly how many zeros follow the 1.

Expression Expanded Value
\(10^1\)
\(10\)
 10
\(10^2\)
\(10 \times 10\)
 100
\(10^3\)
\(10 \times 10 \times 10\)
 1,000
\(10^4\)
\(10 \times 10 \times 10 \times 10\)
 10,000
\(10^6\)
 10 × 10 × ... (6 times) 1,000,000

Common Mistakes to Avoid

✕ Multiplying base by exponent
Writing \( 3^4 = 12 \) instead of \( 3^4 = 81 \). The exponent means repeated multiplication, not regular multiplication.
✕ Distributing the exponent over addition
Writing \( (2+3)^2 = 2^2 + 3^2 = 13 \) instead of \( (2+3)^2 = 5^2 = 25 \). Simplify inside parentheses first.
✕ Forgetting order of operations
Evaluating \( 2 + 3^2 \) as \( 5^2 = 25 \) instead of \( 2 + 9 = 11 \). Exponents apply only to their base, not everything around them.

Quick Reference Table

Expression Expanded Form Value
\( 2^4 \) \( 2 \times 2 \times 2 \times 2 \) 16
\( 3^3 \) \( 3 \times 3 \times 3 \) 27
\( 5^2 \) \( 5 \times 5 \) 25
\( 10^3 \) \( 10 \times 10 \times 10 \) 1,000
\( 7^1 \) \( 7 \) 7
\( 6^0 \) (zero factors) 1

Question 1 of 5

What is the value of \( 6^3 \)?

Question 2 of 5

Which expression equals \( 5 \times 5 \times 5 \times 5 \)?

Question 3 of 5

What is the value of \( 2 + 4^2 \times 3 \)?

Question 4 of 5

What is the value of \( (1 + 4)^2 \)?

Question 5 of 5

Which of the following is not equal to \( 64 \)?