Zero and Negative Indices
What happens when the index is zero? Or negative? Discover the elegant patterns that extend index laws beyond positive whole numbers.
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Worksheet
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Q1 · Looking at the pattern $2^3=8$, $2^2=4$, $2^1=2$, what do you predict $2^0$ equals?
Q2 · What do you think $2^{-1}$ means? Will it be a negative number or something else?
Learning Intentions
Know
- $a^0 = 1$ for any non-zero $a$.
- $a^{-n} = \dfrac{1}{a^n}$ for any non-zero $a$.
Understand
- Why zero and negative indices follow logically from the division index law.
- That negative indices represent reciprocals, not negative numbers.
Can Do
- Evaluate expressions with zero and negative indices.
- Convert between positive and negative index forms.
- Simplify expressions containing mixed positive, zero and negative indices.
Success Criteria
- I can explain why $a^0 = 1$ using the division index law.
- I can write $a^{-n}$ as $\dfrac{1}{a^n}$ and vice versa.
- I can evaluate numerical expressions involving zero and negative indices.
- I can simplify algebraic expressions with negative indices, leaving answers with positive indices.
Key Terms
Common Mistakes to Avoid
Wrong: “$2^{-3} = -8$”. A negative index does NOT mean a negative result. It means the reciprocal.
Right: $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$. The negative tells you to flip to the reciprocal.
Wrong: “$0^0 = 1$”. While $a^0 = 1$ for any non-zero $a$, $0^0$ is undefined (a senior mathematics concept).
Right: $a^0 = 1$ only when $a \neq 0$. Always state this condition.
Why does any non-zero number to the power of zero equal 1? The answer comes from the division index law and a beautiful pattern.
Look at this pattern with base 3:
$3^3 = 27$
$3^2 = 9$ (divide by 3)
$3^1 = 3$ (divide by 3)
$3^0 = 1$ (divide by 3 again)
Using the division law: $\dfrac{3^2}{3^2} = 3^{2-2} = 3^0$. But anything divided by itself equals 1, so $3^0 = 1$.
What to write in your book
- Any non-zero number to the power of zero equals 1: $a^0 = 1$ ($a \neq 0$)
- This follows from the division index law: $a^n / a^n = a^{n-n} = a^0 = 1$
- The pattern $3^3=27$, $3^2=9$, $3^1=3$, $3^0=1$ shows each step divides by 3
- $0^0$ is undefined — always state the condition $a \neq 0$
Negative indices continue the pattern. They do not make the answer negative — they create reciprocals.
Continuing the pattern with base 3:
$3^0 = 1$
$3^{-1} = \dfrac{1}{3}$ (divide by 3)
$3^{-2} = \dfrac{1}{9}$ (divide by 3 again)
$3^{-3} = \dfrac{1}{27}$ (divide by 3 again)
Using the division law: $\dfrac{3^2}{3^5} = 3^{2-5} = 3^{-3}$. Expanded: $\dfrac{3 \times 3}{3 \times 3 \times 3 \times 3 \times 3} = \dfrac{1}{3^3}$. So $3^{-3} = \dfrac{1}{3^3}$.
Key insight: A negative index in the numerator becomes positive in the denominator, and vice versa. Moving a term across the fraction bar changes the sign of its index.
What to write in your book
- A negative index means reciprocal: $a^{-n} = \dfrac{1}{a^n}$ for $a \neq 0$
- A negative index does NOT make the answer negative — it flips the term
- Moving a term across the fraction bar changes the sign of its index
- $\dfrac{1}{a^{-n}} = a^n$ — a negative index in the denominator becomes positive in the numerator
Interactive: Index Transformer
Your Turn
Question 1: Evaluate $5^{-2}$.
Question 2: Simplify $x^{-4} \times x^7$.
Question 3: Write $\dfrac{3}{x^{-2}}$ using positive indices only.
Revisit Your Thinking
Look back at your Think First predictions for $2^0$ and $2^{-1}$. Were they correct? Use the division index law to prove that $2^0 = 1$ and $2^{-1} = \dfrac{1}{2}$. Explain why $2^{-1}$ is not the same as $-2$.
Earlier you were asked: What was your first thought on this topic?
Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?
What is the value of $9^0$?
Evaluate $3^{-2}$.
Simplify $x^{-5} \times x^3$.
Which expression is equivalent to $\dfrac{1}{x^{-3}}$?
Evaluate $\left(\dfrac{2}{3}\right)^{-2}$.
Evaluate the following, giving your answers as fractions where appropriate.
(a) $5^0$ (1 mark)
(b) $2^{-4}$ (1 mark)
(c) $\left(\dfrac{1}{2}\right)^{-3}$ (1 mark)
(a) Simplify $x^{-3} \times x^7$, leaving your answer with a positive index. (2 marks)
(b) Simplify $\dfrac{x^2 \times x^{-5}}{x^{-3}}$, leaving your answer with a positive index. (2 marks)
(a) Explain why $a^0 = 1$ using the division index law. (2 marks)
(b) Write $\dfrac{4x^{-2}}{y^{-3}}$ using only positive indices. (2 marks)
(c) Evaluate $2^{-3} + 2^{-2} + 2^{-1} + 2^0$. (1 mark)
Quick Review
Zero Index
$a^0 = 1$ for any $a \neq 0$
Negative Index
$a^{-n} = \dfrac{1}{a^n}$ for any $a \neq 0$
Reciprocal Rule
$\dfrac{1}{a^{-n}} = a^n$
Across the Fraction Bar
Moving a term across the fraction changes the sign of its index
Not Negative Numbers
$2^{-3} = \dfrac{1}{8}$, not $-8$
Flip Fractions
$\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$
Real-Life Link
Negative indices appear everywhere in science and engineering. In physics, half-life calculations use negative exponents to model radioactive decay. In chemistry, pH is defined using negative powers of 10. In computing, file sizes use prefixes like kilo ($10^3$), mega ($10^6$) and giga ($10^9$), while micro ($10^{-6}$) and nano ($10^{-9}$) describe tiny scales. Understanding negative indices lets you navigate these scales with confidence.
Game Time!
Test your zero and negative index skills in an interactive challenge.
Play Negative Indices Challenge