Evaluating positive fractional indices
Fractional indices help us write roots and powers in a single form, which is useful in areas like science, engineering, and computer graphics where repeated scaling and square roots appear. Learning to evaluate expressions with fractional indices lets you simplify calculations such as square roots, cube roots, and other powers more efficiently. Jump to the questions
Practise now
All answers should be entered as positive values.
Topic guide
What this worksheet practises
This worksheet provides practice on evaluating numbers raised to fractional powers. Understanding fractional indices is essential for higher-level algebra and manipulating surds. A fraction in a power is simply a clever way of writing a root.
Key method
The two parts of a fractional power (the numerator and denominator) do two different jobs. A power in the form a/b means:
- The bottom number 'b' (the denominator) acts as the root. If it's a 2, it means square root. If it's a 3, it means cube root.
- The top number 'a' (the numerator) acts as a standard power. If it's a 1, it changes nothing. If it's a 2, you square the result.
- You can apply the root and the power in either order, but it is almost always easier to apply the root first to make the number smaller before raising it to a power.
Worked example
Evaluate 272/3.
Step 1: Look at the bottom of the fraction. It is a 3. This means we must find the cube root of 27.
³√27 = 3. (Because 3 × 3 × 3 = 27).
Our problem has now simplified to 3².
Step 2: Look at the top of the fraction. It is a 2. This means we must square our new number.
3² = 9.
The final answer is 9.
Common mistakes to avoid
A frequent error is treating the fractional power like a normal multiplication fraction. For instance, calculating 27 × 2/3 (which gives 18). A power is an index, not a multiplier. You must use roots and powers, not standard multiplication.
How to check your answer
If your fractional power is less than 1 (like 1/2 or 2/3), your final answer must be smaller than your starting base number. In our example, 9 is smaller than 27, which proves we used roots rather than accidentally multiplying.