Fractional Indices
[PATHS extension] What if the index is a fraction? Discover the powerful connection between roots and powers that opens the door to advanced algebra.
Printable Worksheets
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Worksheet
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Q1: You know that $9^{1/2} = 3$ because $3^2 = 9$. Before reading further, predict the value of $8^{1/3}$ and $16^{3/4}$. Explain your reasoning for each prediction.
Q2: A student says that $25^{1/2} = 25 \div 2 = 12.5$. Do you agree? Why or why not? What does the fraction $1/2$ as an index actually mean?
Learning Intentions
Know
- $a^{1/n} = \sqrt[n]{a}$
- $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$
Understand
- Why fractional indices connect roots and powers through the index laws.
- That the denominator of the fraction gives the root, and the numerator gives the power.
Can Do
- Evaluate expressions with fractional indices.
- Simplify expressions involving fractional indices.
- Convert between radical form and index form.
Success Criteria
- I can evaluate $a^{1/n}$ by finding the $n$th root of $a$.
- I can evaluate $a^{m/n}$ by finding the $n$th root then raising to the power $m$.
- I can convert between $\sqrt[n]{a}$ and $a^{1/n}$.
- I can simplify expressions using the index laws with fractional indices.
Key Terms
Common Mistakes to Avoid
Wrong: “$8^{1/3} = 8 \div 3$”. A fractional index is NOT division. It means the cube root of 8, which is 2.
Right: $8^{1/3} = \sqrt[3]{8} = 2$ because $2^3 = 8$.
Wrong: “$16^{3/4} = (16^3) \div 4$”. The fraction is an index, not a division problem.
Right: $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$. Denominator gives the root, numerator gives the power.
We have index laws for integer indices. What happens when the index is a fraction? The answer connects two seemingly different operations: roots and powers.
Consider $9^{1/2}$. Using the power of a power law: $(9^{1/2})^2 = 9^{(1/2) \times 2} = 9^1 = 9$. So $9^{1/2}$ is the number that, when squared, gives 9. That is the square root: $9^{1/2} = \sqrt{9} = 3$.
Two ways to evaluate $a^{m/n}$:
Method 1: Find the root first, then apply the power. Usually easier.
Method 2: Apply the power first, then find the root. Gives the same answer.
What to write in your book
- $a^{1/n} = \sqrt[n]{a}$ — the denominator gives the root
- $a^{m/n} = (\sqrt[n]{a})^m$ — numerator gives the power
- Method 1: find the root first, then apply the power (usually easier)
- All index laws apply to fractional indices too
Correct! $27^{1/3} = \sqrt[3]{27} = 3$ because $3^3 = 27$.
Not quite. $27^{1/3} = \sqrt[3]{27} = 3$ because $3^3 = 27$.
Interactive: Root-Index Mapper
Your Turn
Question 1: Evaluate $32^{1/5}$.
Question 2: Evaluate $81^{3/4}$.
Question 3: Write $\sqrt[3]{y^2}$ using a fractional index.
Revisit Your Thinking
Look back at your Think First predictions for $8^{1/3}$ and $16^{3/4}$. Evaluate each expression using the fractional index rules. Were your predictions correct? Explain the connection between roots and fractional indices in your own words.
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?
Evaluate $25^{1/2}$.
Evaluate $64^{2/3}$.
Which expression is equivalent to $\sqrt[4]{x^3}$?
Evaluate $125^{-1/3}$.
Simplify $(x^{1/2})^4$.
Evaluate the following.
(a) $16^{1/4}$ (1 mark)
(b) $27^{2/3}$ (1 mark)
(c) $32^{3/5}$ (1 mark)
(a) Write $\sqrt[5]{x^2}$ using a fractional index. (1 mark)
(b) Write $y^{3/7}$ using a radical (root symbol). (1 mark)
(c) Evaluate $\left(\frac{27}{8}\right)^{2/3}$. (2 marks)
(a) Evaluate $16^{-3/4}$. (2 marks)
(b) Simplify $\dfrac{x^{5/6}}{x^{1/3}}$. (2 marks)
(c) A student claims that $8^{2/3} = (8^2)^{1/3}$. Verify this claim by evaluating both sides. (1 mark)
$a^{1/n}$
$n$th root of $a$
$a^{m/n}$
$(\sqrt[n]{a})^m$
Denominator
The root
Numerator
The power
Negative
Take the reciprocal
Index laws
All apply to fractions too
Real-Life Link
Fractional indices appear in advanced physics and engineering. In Einstein's theory of relativity, the Lorentz factor involves a square root that can be written with a $1/2$ index. In signal processing, fractional Fourier transforms use indices between 0 and 1. While these are senior topics, the foundation you built today — that $a^{m/n}$ connects roots and powers — is the same mathematical idea that appears in university-level science and engineering courses.
Game Time!
Test your fractional index skills in an interactive challenge.
Play Fractional Indices Challenge