Math Lesson 7.1.2 - Properties of Indices

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

Welcome to our Math lesson on Properties of Indices, this is the second lesson of our suite of math lessons covering the topic of Indices, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Properties of Indices

Now, we will explain some properties of indices by illustrating them with numbers as proof.

Property 1: When multiplying powers with the same base, we add the indices.

In symbols,

For example,

Indeed,

Thus,

Property 2: When dividing powers with the same base, we subtract the indices.

In symbols,

For example,

Indeed,

Thus,

Property 3: When raising a power into another power, we multiply the indices without changing the base.

In symbols,

For example,

Indeed,

and

Property 4: If two numbers of different bases are raised at the same power, the bases multiply without changing the index.

In symbols,

For example,

Indeed,

Thus,

Property 5: Any number raised to the first power gives the number itself.

In symbols,

For example,

Indeed, we can write using the second property of indices

Property 6: Any number raised at power zero gives 1.

In symbols,

For example,

Indeed, we can write using the second property of indices

Example 2

Calculate the following expressions using the properties of indices.

1. 12¹ 437/12¹ 435
2. 3⁵ × 3⁷/3⁸

Solution 2

1. From the second property of indices, we have
   12¹ 437/12¹ 435 = 12^{1437 - 1435}
   = 12²
   = 144
   Obviously, it would have been impossible to calculate the result of this expression through the standard way, i.e. calculating the value of numerator and denominator separately and then, dividing them.
2. First, we do the operations in the numerator using the first property of indices. Then, we divide the resulting numerator by the denominator, i.e.
   3⁵ × 3⁷/3⁸ = 3^{5 + 7}/3⁸
   = 3¹ 2/3⁸
   = 3^{12 - 8}
   = 3⁴
   = 81
   Indeed, if we calculated the powers one by one we would obtain
   3⁵ × 3⁷/3⁸ = 243 × 2,187/6,561
   = 531,441/6,561
   = 81

More Indices Lessons and Learning Resources

Whats next?

Enjoy the "Properties of Indices" math lesson? People who liked the "Indices lesson found the following resources useful:

1. Properties Feedback. Helps other - Leave a rating for this properties (see below)
2. Powers and Roots Math tutorial: Indices. Read the Indices math tutorial and build your math knowledge of Powers and Roots
3. Powers and Roots Video tutorial: Indices. Watch or listen to the Indices video tutorial, a useful way to help you revise when travelling to and from school/college
4. Powers and Roots Revision Notes: Indices. Print the notes so you can revise the key points covered in the math tutorial for Indices
5. Powers and Roots Practice Questions: Indices. Test and improve your knowledge of Indices with example questins and answers
6. Check your calculations for Powers and Roots questions with our excellent Powers and Roots calculators which contain full equations and calculations clearly displayed line by line. See the Powers and Roots Calculators by iCalculator™ below.
7. Continuing learning powers and roots - read our next math tutorial: Roots

Help others Learning Math just like you

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

We hope you found this Math tutorial "Indices" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.

Powers and Roots Calculators by iCalculator™

• Expressing Ordinary Numbers In Standard Form Calculator
• Multiplication And Division Of Roots Calculator
• Operations With Standard Numbers Calculator