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# Square roots and fractional powers
- URL: https://www.esheets.io/square-roots-and-fractional-powers/
- Published: 2024-09-12T20:43:39.000Z
- Updated: 2026-06-21T18:10:36.000Z
- Author: Richard Linnington
- Tags: Maths, Number

Square roots and fractional powers are essential for understanding concepts in geometry, physics, and finance. From calculating the side length of a square to determining the rate of growth, these skills help you break down complex problems and solve them efficiently in the real world. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise square roots and fractional powers with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Recognising that a power of 1/2 means square root.
- Evaluating square roots.
- Rewriting fractional powers as roots.
- Simplifying exact values where possible.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

Solve the following square roots using fractional indices. You may provide either the positive or negative root. 

## Topic guide

### What this worksheet practises

This worksheet provides practice on fractional indices (powers that are written as fractions, like 1/2 or 1/3). This is an advanced index law that translates powers into different types of roots.

### Key method

A fractional power is a disguised instruction to find a root.

- Look at the fraction in the power. The **bottom number** (the denominator) tells you what type of root to use.
- A power of **1/2** means the "second root", which is just a normal **Square Root** (√).
- A power of **1/3** means the **Cube Root** (³√).
- A power of **1/4** means the **Fourth Root** (&sup4;√), and so on.
- The top number (the numerator) tells you what normal power to raise the final answer to. If it's a 1 (e.g. 1/2), you just find the root and you're done.

### Worked example

**1) Evaluate 491/2.** 
**2) Evaluate 81/3.**

Example 1: (491/2)

Step 1: The power is 1/2\. The bottom number is 2, so this means "Square Root".

Step 2: Find the square root of 49.

√49 = 7.

The answer is 7.

Example 2: (81/3)

Step 1: The power is 1/3\. The bottom number is 3, so this means "Cube Root".

Step 2: Find the cube root of 8\. (What number multiplied by itself three times makes 8?).

2 × 2 × 2 = 8, so the cube root is 2.

The answer is 2.

### Common mistakes to avoid

The most devastating mistake is treating the fractional power like a normal multiplication. For example, calculating 491/2 as "half of 49" to get 24.5\. This is completely wrong. Powers are not multipliers; they tell you about roots and indices.

### Things to remember

If you encounter a fractional power with a number other than 1 on top, like **82/3**, you do it in two steps. First, use the bottom number to find the root (cube root of 8 is 2). Then, use the top number to power your answer (2 squared is 4). The final answer is 4.