Exponent Rules (Laws of Exponents)

A set of shortcuts for working with powers, all built from what an exponent actually counts.

a^m × a^n = a^m+n

Multiplying powers of the same base just adds the exponents: 3 twos and 2 twos make 5 twos.

What each symbol means

• a: the base (the number being multiplied by itself)
• m, n: the exponents (how many times the base is used)

An exponent is a counting shortcut. 2³ just means 2 × 2 × 2, three twos multiplied together. Every exponent rule below comes straight from that idea, so none of them need memorizing once you see why they are true.

The main rules

• **Product rule:** a^m × a^n = a^m+n (multiply same-base powers by adding the exponents).
• **Quotient rule:** a^ma^n = a^m-n (divide by subtracting the exponents).
• **Power rule:** (a^m)^n = a^m × n (a power of a power multiplies the exponents).
• **Zero exponent:** a^0 = 1 for any nonzero a.
• **Negative exponent:** a^-n = 1a^n (a negative exponent means one over the power).

Same base is the key: The product and quotient rules only work when the base is the same. You can combine 2³ and 2², but not 2³ and 3², because they are made of different building blocks.

A worked example

Simplify 2³ × 2²

1. Both powers have the same base, 2, so use the product rule.
2. Add the exponents: 3 + 2 = 5.
3. So 2³ × 2² = 2^5.
4. Check by counting: (2×2×2)×(2×2) = 2×2×2×2×2 = 2^5 = 32.

• Multiply same-base powers by adding exponents: a^m × a^n = a^m+n.
• Divide by subtracting: a^ma^n = a^m-n.
• A power of a power multiplies: (a^m)^n = a^mn.
• Anything (nonzero) to the power 0 is 1; a negative exponent flips to a fraction.

Why the product rule works

1. Write out what each power means. 2³ = 2 × 2 × 2 (three twos), and 2² = 2 × 2 (two twos).
2. Multiplying them puts all those twos in one long product: 2 × 2 × 2 × 2 × 2.
3. Count the twos: there are 3 + 2 = 5 of them.
4. Five twos multiplied is 2^5. So 2³ × 2² = 2³+2 = 2^5. Adding the exponents is just counting the factors.

Practice questions

Question 1

Simplify 3^4 × 3².

1. Both powers have the same base, 3, so use the product rule.
2. The product rule says multiply same-base powers by adding the exponents.
3. Add the exponents: 4 + 2 = 6.
4. So 3^4 × 3² = 3^6.

Answer: 3^6

Question 2

Simplify 5^75³.

1. Both powers have the same base, 5, so use the quotient rule.
2. The quotient rule says divide same-base powers by subtracting the exponents.
3. Subtract the exponents: 7 - 3 = 4.
4. So 5^75³ = 5^4.

Answer: 5^4

Question 3

Simplify (2³)².

1. This is a power raised to another power, so use the power rule.
2. The power rule says multiply the two exponents.
3. Multiply the exponents: 3 × 2 = 6.
4. So (2³)² = 2^6.

Answer: 2^6

Question 4

What is 9^0?

1. This uses the zero-exponent rule.
2. Any nonzero number raised to the power 0 equals 1.
3. Here the base 9 is not zero.
4. So 9^0 = 1.

Answer: 1

Question 5

Write 2^-3 as a fraction.

1. This uses the negative-exponent rule.
2. A negative exponent means one over the power: 2^-3 = 12³.
3. Work out the bottom: 2³ = 2 × 2 × 2 = 8.
4. So 2^-3 = 18.

Answer: 18