1. Meaning of Exponents

Definition

An exponent tells us how many times to multiply a number, called the base, by itself. The expression aⁿ means "multiply a by itself n times."

• For example, 2⁴ = 2 × 2 × 2 × 2 = 16
• Here, 2 is the base, and 4 is the exponent (or power).

Vocabulary

• Base: The number that is being multiplied.
• Exponent: The number that tells how many times to multiply the base.
• Power: The complete expression, such as 3⁵.

2. Types of Exponents

• Positive Integer Exponents: aⁿ where n is a whole number greater than 0. (Example: 5³ = 125)
• Zero Exponent: Any nonzero number raised to the power 0 is 1. (Example: 7⁰ = 1)
• Negative Exponents: a⁻ⁿ = 1/(aⁿ), a ≠ 0. (Example: 2⁻³ = 1/8)
• Fractional Exponents: a^1/n is the nth root of a. (Example: 27^1/3 = 3)
• Rational Exponents: a^m/n = (nth root of a)^m or (a^m)^1/n.

Special Note

The base a must not be zero for negative or fractional exponents.

3. Laws of Exponents

Exponents follow a specific set of rules that make calculations and simplifications easier. These are called the "laws of exponents."

Product Law

a^m × aⁿ = a^m+n

Quotient Law

a^m ÷ aⁿ = a^m−n (a ≠ 0)

Power of a Power Law

(a^m)ⁿ = a^m×n

Product to a Power Law

(ab)ⁿ = aⁿ × bⁿ

Quotient to a Power Law

(a/b)ⁿ = aⁿ ÷ bⁿ (b ≠ 0)

Zero Exponent Law

a⁰ = 1 (a ≠ 0)

Negative Exponent Law

a⁻ⁿ = 1/(aⁿ) (a ≠ 0)

Examples

• 2³ × 2⁵ = 2⁸ = 256
• 7⁵ ÷ 7² = 7³ = 343
• (3²)⁴ = 3⁸ = 6,561
• (4 × 5)² = 4² × 5² = 16 × 25 = 400
• (9/3)² = 9² ÷ 3² = 81 ÷ 9 = 9
• 10⁰ = 1
• 5⁻² = 1/25

4. Working with Large and Small Numbers

Exponents make it easy to write very big or very small numbers. Instead of writing out all the zeros, use powers of 10.

Standard Form (Scientific Notation)

• A number in standard form is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
• For example, 3,500,000 = 3.5 × 10⁶
• Another example, 0.00027 = 2.7 × 10⁻⁴

Converting to Standard Form

• Move the decimal point to get a number between 1 and 10. Count the number of places you move; this is n.
• If the original number is large, n is positive. If it is small (less than 1), n is negative.

5. Comparing Numbers with Exponents

When numbers are written with exponents, compare them by rewriting them with the same base or by calculating their values.

• Which is greater: 2⁵ or 3⁴?
  2⁵ = 32, 3⁴ = 81, so 3⁴ is greater.
• Which is smaller: 5⁻² or 2⁻³?
  5⁻² = 1/25 = 0.04, 2⁻³ = 1/8 = 0.125, so 5⁻² is smaller.

6. Applications of Exponents and Powers

• Area and Volume Calculations: Area of a square is a², volume of a cube is a³.
• Scientific Notation: Useful for representing the distance from Earth to the Sun or the size of atoms.
• Computer Science: Memory and data sizes (kilobytes, megabytes, etc.) use powers of 2 and 10.
• Finance: Compound interest formulas involve exponents: A = P(1 + r/100)ⁿ
• Population Growth: Exponential growth or decay in biology, physics, and chemistry.

Real-World Example

The speed of light is about 300,000,000 meters per second. In standard form, that is 3 × 10⁸ m/s.

The mass of a proton is about 0.00000000000000000000000167 kg. In standard form, that is 1.67 × 10⁻²⁷ kg.

7. Powers of Ten

Understanding powers of ten makes it easy to multiply and divide by 10, 100, 1000, and so on.

• 10¹ = 10
• 10² = 100
• 10³ = 1,000
• 10⁻¹ = 0.1
• 10⁻² = 0.01
• 10⁻³ = 0.001

Using Powers of Ten in Calculations

• Multiplying a number by 10ⁿ moves the decimal n places to the right.
• Dividing by 10ⁿ moves the decimal n places to the left.

Example: 4.5 × 10³ = 4,500; 6.72 × 10⁻² = 0.0672

8. Multiplying and Dividing with Exponents

Multiplication Example

(2³) × (2⁴) = 2³⁺⁴ = 2⁷ = 128

Division Example

(8⁵) ÷ (8²) = 8⁵⁻² = 8³ = 512

Power to a Power Example

(5²)⁴ = 5^2×4 = 5⁸ = 390,625

9. Roots and Fractional Powers

Exponents can represent roots as well as repeated multiplication.

• a^1/2 = √a (square root of a)
• a^1/3 = ∛a (cube root of a)
• a^m/n = (nth root of a)^m

Example

16^1/2 = √16 = 4
27^1/3 = ∛27 = 3
81^3/4 = (4th root of 81)³ = (3)³ = 27

10. Common Errors and How to Avoid Them

• Forgetting to apply laws of exponents correctly. Always combine exponents when multiplying or dividing with the same base.
• Incorrectly applying zero exponent law to zero. Remember, 0⁰ is undefined; a⁰ is 1 only when a ≠ 0.
• Using negative bases incorrectly. For example, (−3)² = 9, but −3² = −9.
• Mixing up order of operations (PEMDAS/BODMAS). Handle exponents before multiplication or addition.
• Confusing product law and power of a power law. Practice with several examples to build accuracy.

11. Practice Problems

Basic

1. Write 81 as a power of 3.
2. Evaluate: 2⁵ × 2³
3. Find the value of 10⁻³
4. Simplify: (4²)³
5. Write 0.0004 in standard form.

Intermediate

6. Express 125 as a power of 5.
7. Simplify: (3⁴ × 3²)/3³
8. If a² = 49, find the value of a.
9. Simplify: 5⁰ + 2⁻²
10. Find the cube root of 64 using exponents.

Advanced

11. Simplify: (2³ × 4²)/8
12. Evaluate: (5³)² × 5⁻⁴
13. Convert 2,300,000 to standard form.
14. Simplify: (x⁵ × x⁻²)/x³
15. If (2^x = 16), what is the value of x?

12. Solutions

1. 81 = 3⁴
2. 2⁵ × 2³ = 2⁸ = 256
3. 10⁻³ = 1/1,000 = 0.001
4. (4²)³ = 4⁶ = 4,096
5. 0.0004 = 4 × 10⁻⁴
6. 125 = 5³
7. (3⁴ × 3²)/3³ = 3⁶⁻³ = 3³ = 27
8. a² = 49 ⇒ a = 7 or a = −7
9. 5⁰ + 2⁻² = 1 + 1/4 = 1.25
10. Cube root of 64 = 64^1/3 = 4
11. (2³ × 4²)/8 = (8 × 16)/8 = 128/8 = 16
12. (5³)² × 5⁻⁴ = 5⁶⁻⁴ = 5² = 25
13. 2,300,000 = 2.3 × 10⁶
14. (x⁵ × x⁻²)/x³ = x⁵⁻²⁻³ = x⁰ = 1
15. 2^x = 16 ⇒ 16 = 2⁴ ⇒ x = 4

Conclusion

Exponents and powers unlock the ability to handle numbers that are too large or small for ordinary calculations. They are essential tools in advanced mathematics, science, computing, and many fields of research and daily life. By practicing these concepts, you build a strong foundation for future study and problem solving. Remember to use the laws of exponents accurately, practice expressing numbers in standard form, and always check your work for errors.