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.
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.
8 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.