##### How many solutions?

Use graphs and the nsolve(...) command to explore equations' solutions and discover properties of negative indices.

Publisher: Texas Instruments UK

Editor: Barrie Galpin

Author: Nevil Hopley

Topic: Mathematics

Tags Decimals , Exponential , Powers and roots , Calculator methods , Index , Mathematical thinking

Students will learn how to use the nsolve(...) command to find solutions to equations that can only be tackled with numerical methods. They will also encounter the 'with' syntax command (the **|** symbol), and how this can be used to help search for, and check, particular solutions.

The equations are graphed in two different ways, and function tables are used to give further insight where needed.

All in all, this activity develops good investigational skills through multiple representations, that can lead students to confidently explore index notation with powers of 0, -1, -2, etc. on their own.

A detailed 8-page **Handout** guides students, step by step, through the necessary constructions starting with a new, blank TI-Nspire document. There is also a **Marking Grid**, which will be helpful in assessing students? completed work.

The Activity is divided into 4 tasks:

1. How many solutions to x^{2}=2^{x}?

2. How many solutions to x^{3}=3^{x}?

3. How many solutions to x^{n}=n^{x},^{ }when n is a positive integer?

4. What if n is a negative integer?

Even if students don't progress beyond Task 3 of the activity, they will have gained good practice at using key in-built features of the TI-Nspire handheld.

(NB. The student handout includes screenshots and Calculator Application messages produced using TI-Nspire version OS3.x)