Extending the power tower downward (base 2)
- 2⁰ = 2 ÷ 2 = 1
- 2⁻¹ = 1 ÷ 2 = ½
Year 9 maths lesson · Indices and Square Roots
Zero and Negative Powers
Explore what happens when the exponent is zero or negative. Discover that x⁰ = 1 for any base, and that negative powers represent repeated division — so x⁻ⁿ = 1/xⁿ.
- About 4–6 minutes
- 10 quick questions as you go
- Then about 6 minutes of practice
The lesson didn't load. Refresh the page to try again.
Worked examples
Base 3 tower extended
- 3⁰ = 3 ÷ 3 = 1
- 3⁻¹ = 1 ÷ 3 = ⅓
- 3⁻² = ⅓ ÷ 3 = 1/9
Applying the rule: 2⁻³
- Negative power means: 1 divided by the base, repeated
- 2⁻³ = 1/2³ = 1/8
More examples
- 5⁻² = 1/5² = 1/25
- 10⁻³ = 1/10³ = 1/1000
Before this lesson
It helps to be comfortable with:
- Expanded and index form
- Division of whole numbers
Lesson transcript
Everything the lesson says, word for word
Student: Hey Alex, what’s that on the board? 2 to the negative 3?
Student: I bet I could work that out… powers mean repeated multiplication, right?
Student: Like, 2 cubed is 2 times 2 times 2, which is 8.
Student: But you can’t multiply something a negative number of times… that doesn’t make sense.
Teacher: Good thinking. Let’s look at some patterns and see if we can figure it out.
Teacher: Let’s start with what we know. 2 to the power of 1 is just 2.
Teacher: 2 squared is 2 times 2, that’s 4.
Teacher: And 2 cubed is 2 times 2 times 2, which is 8.
Teacher: See the pattern? Each time we go up a power, we multiply by 2.
Teacher: But what if we go the other way — down?
Student: If going up means multiply by 2… then going down must mean divide by 2!
Teacher: So what’s 2 to the power of 0?
Student: 2 to the power of 1 is 2, divide by 2… that’s 1! So 2 to the power of 0 equals 1!
Student: And 2 to the negative 1 would be… 1 divided by 2, which is a half!
Teacher: Exactly. And we can do the same with any base. Let’s try 3.
Teacher: Here we divide by 3 to go down. So 3 to the power of 0 is still 1, and 3 to the negative 1 is a third.
Teacher: And if we divide by 3 again, 3 to the negative 2 is 1 over 9.
Student: Ooh, I notice something! If I use this row as a reference point… the negative exponent tells you how many times to divide by 3. Like 3 to the negative 2 — you start at 1 and divide by 3 twice!
Teacher: So anything to the power of 0 is 1. And a negative power means repeated division of 1.
Student: So going back to 2 to the negative 3… That’s 1 divided by 2, three times. That’s 1 over 2 cubed. Which is 1 over 8!
Teacher: Let’s try a couple more. What’s 5 to the negative 2?
Student: That’s 1 divided by 5, two times. So 1 over 5 squared, which is 1 over 25!
Teacher: Nice. And 10 to the negative 3?
Student: 1 divided by 10, three times. That’s 1 over 10 cubed, which is 1 over 1000!
Teacher: Right, try these questions.
More Indices and Square Roots lessons
- Expanded Form and Index FormConvert between expanded form and index (exponent) notation.
- Powers of 2 and 10Learn the powers of 2 and 10 and their importance.
- Estimating Square RootsUnderstand square roots and estimate roots of non-perfect squares.
Next lesson: Estimating Square Roots →All Year 9 Number lessonsAll lessons