Fractional Indices and Roots
Read x^(1/n) as the nth root and x^(m/n) as a root and a power together, and move between root form and index form.
What a learner can do afterwards
- Work out 27^(1/3) and 16^(3/4) without a calculator
- Rewrite a root as a fractional index and back again
- Handle a negative fractional index in one calculation
1 · Read
x to the power 1/n means the nth root. 27^(1/3) asks which number multiplied by itself three times gives 27, and that number is 3. A power of 1/2 is the square root and 1/5 is the fifth root, so 32^(1/5) = 2.
For x^(m/n), take the root first, then the power. 16^(3/4): the fourth root of 16 is 2, then 2 cubed is 8. Root first keeps the numbers small. 8^(2/3) works the same way: cube root 2, then squared, giving 4.
Roots and indices are the same idea in two outfits, and you can swap them. The cube root of x^5 becomes x^(5/3): inside power on top, root type below. Back the other way, x^(2/5) is the fifth root of x squared.
A minus sign only means flip to the reciprocal, and you already know that move. For 27^(-1/3), flip first to 1 over the cube root of 27, which is 1/3. Do the flip, then the root and power, one step at a time.
Bottom of the fraction picks the root, top picks the power, minus picks the flip.
More to read
How to Simplify Square Roots
Square roots aren't just a button on your calculator. You can work them out by hand if you break the number apart first.
Start by splitting the number under the root sign into its prime factors. Draw a factor tree: keep splitting the number into smaller pieces until every piece is a prime number (a number only divisible by 1 and itself).
Once you have the prime factors, look for pairs. Every matching pair of primes can come out of the square root as a single copy of that number. Any prime left without a partner has to stay under the root sign.
This same idea of breaking a number down is exactly what you use when you work out something like 27^(1/3) by hand: instead of guessing, you think about what number multiplied by itself the right number of times gives you the number you started with.
Practicing this by hand first makes it much easier to understand what a fractional index is really asking for, before you ever reach for a calculator.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
20 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.