Exponents - Seventh Grade

Powers, Bases & Scientific Notation

1. Understanding Exponents

Definition

bⁿ

b = Base (number being multiplied)

n = Exponent/Power (how many times)

An exponent tells how many times to multiply the base by itself

• Read as "b to the power of n" or "b to the nth power"

• Special names: b² = "b squared", b³ = "b cubed"

Examples

Exponential Form	Expanded Form	Value
2³	2 × 2 × 2	8
5²	5 × 5	25
3⁴	3 × 3 × 3 × 3	81

2. Evaluating Powers

Steps to Evaluate

Step 1: Identify the base and exponent

Step 2: Multiply the base by itself (exponent) times

Step 3: Calculate the result

Special Exponent Rules

b¹ = b

Any number to the power of 1 equals itself

b⁰ = 1

Any number (except 0) to the power of 0 equals 1

Examples

4³ = 4 × 4 × 4 = 64

10² = 10 × 10 = 100

7¹ = 7

12⁰ = 1

3. Powers with Negative Bases

Sign Rules

EVEN exponent → POSITIVE result

(-base)^(even) = Positive

Example: (-2)⁴ = 16

ODD exponent → NEGATIVE result

(-base)^(odd) = Negative

Example: (-2)³ = -8

Examples

Expression	Expanded	Result
(-3)²	(-3) × (-3)	9 (positive)
(-3)³	(-3) × (-3) × (-3)	-27 (negative)
(-2)⁴	(-2) × (-2) × (-2) × (-2)	16 (positive)
(-5)³	(-5) × (-5) × (-5)	-125 (negative)

4. Powers with Decimal and Fractional Bases

Decimal Bases

Multiply the decimal by itself (exponent) times

Example 1: (0.5)²

(0.5)² = 0.5 × 0.5 = 0.25

Example 2: (1.2)³

(1.2)³ = 1.2 × 1.2 × 1.2 = 1.728

Fractional Bases

(a/b)ⁿ = aⁿ/bⁿ

Apply exponent to BOTH numerator and denominator

Example: (2/3)³

(2/3)³ = 2³/3³ = 8/27

5. Powers of Ten

Positive Powers of 10

10ⁿ = 1 followed by n zeros

Power	Expanded	Value
10¹	10	10
10²	10 × 10	100
10³	10 × 10 × 10	1,000
10⁶	10 × 10 × 10 × 10 × 10 × 10	1,000,000

Negative Powers of 10

10⁻ⁿ = 1/(10ⁿ) = 0.000...1

Decimal point moves n places to the left

Power	Fraction	Value
10⁻¹	1/10	0.1
10⁻²	1/100	0.01
10⁻³	1/1,000	0.001

6. Scientific Notation

Definition

Scientific notation expresses numbers as:

a × 10ⁿ

Where: 1 ≤ a < 10

And: n is an integer

Rules for Scientific Notation

Large Numbers (≥ 10):

• Move decimal LEFT → Positive exponent

• Count places moved = exponent value

Small Numbers (< 1):

• Move decimal RIGHT → Negative exponent

• Count places moved = exponent value

Examples

Standard Form	Scientific Notation
5,000	5 × 10³
350,000	3.5 × 10⁵
0.006	6 × 10⁻³
0.000025	2.5 × 10⁻⁵

Converting TO Scientific Notation

Example 1: Convert 42,000 to scientific notation

Move decimal 4 places left: 4.2

Exponent: 4 (positive because number is large)

Answer: 4.2 × 10⁴

Example 2: Convert 0.0078 to scientific notation

Move decimal 3 places right: 7.8

Exponent: -3 (negative because number is small)

Answer: 7.8 × 10⁻³

7. Comparing Numbers in Scientific Notation

Steps to Compare

Step 1: Compare the EXPONENTS first

• Larger exponent = Larger number

Step 2: If exponents are EQUAL, compare the coefficients

• Larger coefficient = Larger number

Examples

Example 1: Compare 3.5 × 10⁵ and 2.8 × 10⁷

Exponents: 5 vs 7

7 > 5, so 2.8 × 10⁷ is larger

Answer: 2.8 × 10⁷ > 3.5 × 10⁵

Example 2: Compare 4.2 × 10⁴ and 6.1 × 10⁴

Exponents are equal: both 10⁴

Compare coefficients: 6.1 > 4.2

Answer: 6.1 × 10⁴ > 4.2 × 10⁴

8. Evaluating Expressions with Exponents

Order of Operations (PEMDAS)

Remember: PEMDAS

Parentheses

Exponents ← Do these SECOND!

Multiplication & Division (left to right)

Addition & Subtraction (left to right)

Example

Evaluate: 2 + 3² × 4

Step 1: Evaluate exponent: 3² = 9

Expression becomes: 2 + 9 × 4

Step 2: Multiply: 9 × 4 = 36

Expression becomes: 2 + 36

Step 3: Add: 2 + 36 = 38

Answer: 38

Quick Reference: Exponent Rules

Rule	Formula/Example
Definition	bⁿ = b × b × b... (n times)
Power of 1	b¹ = b
Power of 0	b⁰ = 1
Negative Base (Even)	(-b)^(even) = Positive
Negative Base (Odd)	(-b)^(odd) = Negative
Fractional Base	(a/b)ⁿ = aⁿ/bⁿ
Powers of 10 (Positive)	10ⁿ = 1 followed by n zeros
Powers of 10 (Negative)	10⁻ⁿ = 1/(10ⁿ) = 0.000...1
Scientific Notation	a × 10ⁿ (where 1 ≤ a < 10)