Solving Equations with Indices

June 1, 2022

Differentiated Learning Objectives

All students should use the power rule to solve equations with indices of the form a^x = (a^b)^x
Most students should find a common base and use the power rule to solve equations.
Some students should find a common base and use the power rule to solve equations involving fractions.

Links to Lesson Resources (Members Only)

Starter/Introduction

At the start of the lesson, students recap using the power and multiplication rules of indices to simplify four products.

Students may need reminding how to use the power rule to change the base of a number. For instance, we write 8 as 2³, and 1/4 equals 2^-2.

Students should use mini whiteboards to match the blue card with its correct pink card. It is essential to feedback on the solutions as students will need this skill throughout the lesson. Those who finish early could check their solutions on a calculator.

Prompts / Questions to consider

Do 8 and 4 have a common base?
How can we write 8 and 4 use the power of 2?
How can a negative power help us write 1/4 using the base of 2?

Solving Equations with Indices

To introduce solving equations with indices, ask the class to discuss in pairs or small groups how the equation 2^x+1= 4^x is different to the types of equations they have seen before. A typical response is that the unknown is now forms a power.

To solve the equation, we need to remember that the equal sign means the two sides of the equation are equal. So if we can write the two bases, 2 and 4, with the same base, the only difference in the two sides will be their powers. Next, ask the class if there is a way to write 2 and 4 with the same base. A typical response is 4 = 2².

Having worked through the first example, students could now work in pairs on a single whiteboard to solve the equation below.

4^{x+1}=8^x

Review the student’s work on their mini-whiteboards and give feedback by modelling the correct approach.

Work through the remainder of the problems on the second slide as shown in the video below.

When working through the questions on the third slide, students should state each stage of their working including:

Any change of base
Use of the power rule to equate the powers
Using the balance method to solve the final equation.

Plenary

The plenary challenges students to combine indices’ power, multiplication, and division rules. Encourage students to start by writing 4, 8 and 32 with a common base. Encourage students to share their approaches with each other to develop their reasoning.

This activity takes between 8 and 10 minutes for students to complete confidently. Students who finish early could check their solution by substituting x back into the equation.

Prompts / Questions to consider

How can we write 4, 8 and 32 using the same base?
Can the denominator be moved to the right-hand side of the equation?
How can we check the solution?

Differentiation

To challenge the more able students, students could solve equations involving roots and fractional powers. Less able students may benefit from having a list of the first three positive and negative powers of 2, 3 4 and 5 as this will aid their processing time.