# Powers and Roots

Students are required to be fluent in the use of numbers in index form and surd form. This resource list contains activities designed for students to use integer powers and associated roots; square, cube and higher, including the use of surd notation e.g. √8, and distinguish between exact answers and decimal approximations.

Visit the secondary mathematics webpage to access all list.

### Powers and Roots

Powers and roots pack one contains an activity explaining why square numbers are called square numbers and continues by requiring students to estimate square roots using trial and improvement to refine their answer.

**Power Match** requires students to pair up numbers expressed as powers and their evaluated form.

Powers and roots pack two contains questions covering the index laws. **Paper Power** explians why anything to the power 0 is equal to 1 by cutting a piece of paper in half multiple times.

### Focus Year 7/8 Number Extension

**Powers** beginning on page 74 (page 76 of the pdf) begins with a dice game called squirm which is an introduction to the concept of powers and the notation used. Square tables are investigated and the notion of a perfect square. The section contains a variety of ideas including how to find a quick way to find the difference of two squares. Cube numbers are discussed before moving on to how powers can be multiplied and divided.

### Indices and Standard Form

**Indices and Standard Form ** introduces index notation and writing numbers using index notation. There is a nice description of what x to the power of n means which is useful to start this topic off.

**The laws of indices** contains explanations, examples and exercises including a formal proof of the rules of indices and an explanation of why x^0 = 1. Students could be asked to investigate the laws for themselves.

**Negative indices** shows that a to the power -1 means the same as 1/a. The term reciprocal could be introduced here. It would be useful to recap arithemetic with negative numbers as a starter since some questions require students to perform calculations such as 4 - (- 3).

**Fractional indices** link to roots and powers. There is a useful 'true or false' exercise in question 5, which can be used to highlight common misconceptions and could be used as a mini-plenary to check students' understanding.

### Using Indices N12

Students are introduced to** fractional **and **negative **indices and require knowledge of the rules of indices for multiplying and dividing numbers in index form.

The pack contains two sets of matching cards, one with numbers and one with algebra. The resource contains a group activity in which students match pairs of cards. The task can be differentiated by providing questions and answers on different coloured card or by starting with the cards face up. This could be used as an extension activity for more able students.

### Arithmetic

This resource contains a number of activities that can be used as inspiration when planning. Appropriate resources for this topic are:

**Powers and Roots: **an activity that starts by introducing powers and continues to cover the rationalising of fractional surds.

**What is a surd?: **This resource provides an introduction to surds and highlights that √(a + b) does not always equal √a + √b. Examples of the use of surds are provided using Pythagoras' Theorem and the quadratic formula.

This resource could be used as inspiration for a lesson to highlight that calculations using numbers in surd form are less prone to rounding errors.

### Awkward Triangle

In this resource from the 'Something in Common' collection students are given a series of triangles with lengths in surd form. Can students find the area without using a calculator? All of the questions are different, but the solutions all have something in common.

To find the solution students practice using the cosine rule, sine rule and manipulation of surds.