Powers and Exponents Study Guide

Introduction to Powers and Exponents

We must say that there are as many squares as there are numbers.

–Galileo Galilei (1564–1642)

This lesson will uncover important properties of powers and exponents. You will discover how to simplify and evaluate various types of exponents.

How do you raise numbers to different powers? Well, let's look at an example. In 3², we call 3 the base and 2 the exponent.

3² = 3 × 3 = 9

When you raise integers to different powers, it is important to remember the following rules:

• Raising a number greater than 1 to a power greater than 1 results in a bigger number: 2² = 4
• Raising a fraction between 0 and 1 to a power greater than 1 results in a smaller number:
• A positive or negative integer raised to the power of 0 is always equal to 1: 9⁰ = 1
• A positive integer raised to any power is equal to a positive number: 9² = 9 × 9 = 81
• A negative integer raised to any even power is equal to a positive number: (–8)² = –8 × –8 = 64
• A negative integer raised to any odd power is equal to a negative number: (–3)³ = –3 × –3 × –3 = 9 × –3 = –27
• A positive or negative integer raised to a negative power is always less than 1:

Think of exponents as a shorthand way of writing math. Instead of writing 2 × 2 × 2 × 2, you can write 2⁴. This saves time and energy—plus, it means the same thing!

Operations and Exponents

When you multiply powers of the same base, add the exponents:

2² × 2³ = 2^{2 + 3} = 2⁵

Think of it this way: 2² × 2³ = (2 × 2) × (2 × 2 × 2), which is the same as 2⁵.

When you divide powers of the same base, subtract the exponents:

3⁸ ÷3² = 3^{8 – 2} = 3⁶

When you raise a power to a power, multiply the exponents:

(5³)⁴ = 5^{3 · 4} = 5¹²

Bottom line: If you're in doubt when multiplying or dividing powers, expand it out! Not sure about 2² × 2⁴? Rewrite it as (2 × 2)(2 × 2 × 2 × 2), which is 2 × 2 × 2 × 2 × 2 × 2, or 2⁶.

When there are multiple bases inside parentheses that are being raised to a power, each base in the parentheses must be evaluated with that power. Try to simplify the following:

(2x²y³)³

2³x² · ³y³ · ³

8x⁶y⁹

The exponent outside the parentheses was evaluated on each base—2, x, and y.

Different Bases

When you do not have the same base, try to convert to the same base:

25⁴ × 5¹² = (5²) ⁴ × 5¹² = 5⁸ × 5¹² = 5²⁰

Try solving for x in the following equation:

2^{x + 2} = 8³

First, get the base numbers equal. Because 8 can be expressed as 2³, then 8³ = (2³)³ = 2⁹. Both sides of the equation have a common base of 2, 2^{x + 2} = 2⁹, so set the exponents equal to each other to solve for x: x + 2 = 9. So, x = 7.

Negative Exponents

Any base number raised to a negative exponent is the reciprocal of the base raised to a positive exponent. Confused? Let's look at a real example:

When simplifying with negative exponents, remember that .

Find practice problems and solutions for these concepts at Powers and Exponents Practice Questions.