Properties of integer exponents · lesson

Zero and negative exponents

💡 Any nonzero number to the zero power is 1, and a negative exponent means one over the positive power.

Each step down in the exponent divides by the base. 2³ = 8, 2² = 4, 2¹ = 2, so 2⁰ = 1. Any nonzero number to the 0 power is 1.

Keep going down and you get fractions. 2⁻¹ = 1/2 and 2⁻² = 1/4. A negative exponent means one over the positive power.

Powers of 10 with negative exponents are small decimals. 10⁻⁴ = 1/10,000 = 0.0001.

🧠 Let's work it out

Find 2⁻⁵.

What do you think comes next?

exponent: the small raised number that tells how many times to use the base as a factor
base: the number being multiplied, like 2 in 2⁵
power: a base with an exponent, like 2⁵

💡 When you multiply powers with the same base, add the exponents.

x⁴ · x² means (x · x · x · x)(x · x). That is six x's, so it is x⁶.

The rule: same base, multiply, add the exponents. It works with negative exponents too.

🧠 Let's work it out

Find 3² × 3⁻⁵.

What do you think comes next?

💡 When you divide powers with the same base, subtract the exponents.

3⁷ ÷ 3⁴ means seven 3's on top and four on the bottom. Four pairs cancel, leaving three 3's.

The rule: same base, divide, subtract the bottom exponent from the top one.

🧠 Let's work it out

Find 3⁷ ÷ 3⁴.

What do you think comes next?

💡 When you raise a power to another power, multiply the exponents.

(2²)³ means 2² · 2² · 2². That is 2 to the 2 + 2 + 2 = 6 power.

The rule: a power raised to a power, multiply the exponents.

For a long problem, simplify inside the parentheses first. Then use the rules one at a time.

🧠 Let's work it out

Find (2⁻²)².

What do you think comes next?