Step 1
Express both sides with the same base.
🎯 Learning Objectives
- simplify expressions involving several index laws;
- write answers with positive indices;
- solve index equations by expressing both sides with the same base.
Introduction
This lesson develops Simplifying & Index Equations from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Visual Exploration
Notice→Connect→Explain
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
✏️ Simplifying Expressions
Simplify (3x 4 y −2 )(2x 3 y 5 ) .
Multiply coefficients:
3×2=6
.
Add indices of
x
:
x
4+3
=x
7
.
Add indices of
y
:
y
−2+5
=y
3
.
Answer:
6x
7
y
3
.
Solving Index Equations
When possible, rewrite both sides using the same base. Then equate the indices.
2
x+1
=16=2
4
⇒ x+1=4 ⇒ x=3
Step 2
Equate indices because equal powers of the same positive base have equal exponents.
Step 3
Solve the resulting linear or quadratic equation.
✏️ A harder example
Solve 9 x =27 .
Use base 3:
9=3
2
and
27=3
3
.
(3
2
)
x
=3
3
, so
3
2x
=3
3
.
2x=3
, hence
x=3/2
.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
- Solve 5 x =125 .
- Solve 2 2x−1 =32 .
- Solve 27 x =9 .
- Simplify (a 5 b −2 )/(a 2 b 3 ) using positive indices.
Show answers
- x=3
- x=3
- x=2/3
- a 3 /b 5
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.