Exponents and Powers — Rules and Examples
Exponents are a shorthand way to express repeated multiplication. Whether you're simplifying algebraic expressions, working with scientific notation, or solving equations, understanding exponent rules is essential. This guide covers every rule you need, with clear examples at each step.
What Are Exponents?
An exponent tells you how many times to multiply a number by itself. In the expression 23, the number 2 is the base and the number 3 is the exponent (also called the power). It means 2 × 2 × 2, which equals 8.
You can read 23 as “two to the third power” or “two cubed.” Similarly, 52 is “five squared” and equals 5 × 5 = 25. Exponents let you write very large (or very small) numbers compactly and follow consistent rules that make calculations much easier.
Basic Exponent Rules
The three foundational rules govern how exponents behave during multiplication, division, and when a power is raised to another power. Each rule only applies when the bases are the same.
Product Rule
When you multiply two powers with the same base, add the exponents: am × an = am+n.
Example: 23 × 24
Add the exponents: 23+4 = 27. Calculate: 27 = 128.
Quotient Rule
When you divide two powers with the same base, subtract the exponents: am ÷ an = am−n.
Example: 56 ÷ 52
Subtract the exponents: 56−2 = 54. Calculate: 54 = 625.
Power Rule
When you raise a power to another power, multiply the exponents: (am)n = am×n.
Example: (32)3
Multiply the exponents: 32×3 = 36. Calculate: 36 = 729.
Zero and Negative Exponents
Zero exponent: Any non-zero number raised to the power of zero equals 1. Why? Consider the pattern: 23 = 8, 22 = 4, 21 = 2 — each time the exponent decreases by one, you divide by 2. Following that pattern, 20 = 2 ÷ 2 = 1. This holds for every non-zero base: a0 = 1.
Negative exponent: A negative exponent means “take the reciprocal.” The rule is a−n = 1/an. Instead of multiplying, you divide — the negative sign flips the base to the denominator.
Example: 2−3
Apply the rule: 2−3 = 1/23 = 1/8 = 0.125.
Example: 10−2
Apply the rule: 10−2 = 1/102 = 1/100 = 0.01.
Power of a Product and Quotient
When a product is raised to a power, distribute the exponent to each factor: (ab)n = an × bn. The same logic applies to quotients: (a/b)n = an / bn.
Example: (2 × 3)2
Distribute the exponent: 22 × 32 = 4 × 9 = 36. You can verify: 62 = 36.
Example: (3/4)2
Distribute the exponent: 32 / 42 = 9/16.
Scientific Notation
Scientific notation uses exponents to write very large or very small numbers in a compact form. The format is a × 10n, where a is a number between 1 and 10 (1 ≤ a < 10) and n is an integer exponent.
To convert a large number, move the decimal point left until you have a single digit before it, and count the moves — that count becomes the exponent. For small numbers, move the decimal point right and use a negative exponent.
Example: 3,200,000 in scientific notation
Move the decimal 6 places left → 3.2 × 106.
Example: 0.00047 in scientific notation
Move the decimal 4 places right → 4.7 × 10−4.
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