Exponents and Powers

Exponents are a way to represent repeated multiplication of the same number. The exponent tells us how many times to use the base as a factor.

1. What is an Exponent?

In the expression aⁿ, a is called the base and n is called the exponent or power. It means a is multiplied by itself n times.

2. Laws of Exponents

• Product of Powers: a^m × aⁿ = a^m+n
• Quotient of Powers: a^m ÷ aⁿ = a^m-n
• Power of a Power: (a^m)ⁿ = a^m×n
• Power of a Product: (ab)ⁿ = aⁿ × bⁿ
• Power of a Quotient: (a/b)ⁿ = aⁿ / bⁿ
• Zero Exponent: a⁰ = 1 (where a ≠ 0)
• Negative Exponent: a^-n = 1 / aⁿ (where a ≠ 0)

3. Standard Form

Very large or very small numbers can be written using exponents, called standard form or scientific notation.
Example: 3,000,000 = 3 × 10⁶

4. Comparing Numbers Using Exponents

Exponents make it easy to compare large numbers.
Example: 2⁵ = 32, 2⁸ = 256, so 2⁸ > 2⁵

5. Word Problems

1. Question: Express 81 as a power of 3.
   Solution: 81 = 3 × 3 × 3 × 3 = 3⁴
2. Question: Simplify: 5³ × 5².
   Solution: 5³⁺² = 5⁵ = 3125
3. Question: Write 1/125 as a power of 5.
   Solution: 1/125 = 5^-3

6. Practice Exercises

• Write 64 as a power of 2.
• Simplify: 7² × 7³
• Express 1/16 as a power of 2.
• Write 10,000 in standard form.
• Simplify: (3²)³

7. Summary

• Exponents represent repeated multiplication.
• There are several laws of exponents to simplify expressions.
• Standard form helps write very large or small numbers easily.
• Negative and zero exponents have special meanings.