How To Teach The Quotient Rule Of Exponents

Key Concepts of Quotient Rule of Exponents

It is important for students to discover the rules of exponents by doing an exploratory activity instead of memorizing the rules. It is much more intuitive for students to divide the exponents when dividing powers. An exploratory activity helps students understand why it makes sense to subtract the powers instead. In this blog, we talk about how you can teach students about the quotient rule.

How do you teach the quotient rule of exponents?

Step 1: Expand the expressions

Have students complete the following table independently.

When it looks like students are done, ask a student or multiple students to fill in the table on the board. 

Step 2: Cancel common factors

Review with students that anything divided by itself is equal to 1. You can show them an example of 2/2 = 1. Then have students cancel out any common factors in their expanded form and circle any factors that are leftover.

When it looks like students are done, ask a student or multiple students to cancel the common factors and circle the leftover factors on the board.

Also Read: How To Teach Zero And Negative Exponents

Step 3: Write the simplified exponent form

Then have students write in the simplified exponent form for each example.

Step 4: Discussion

“Using the table, can we come up with a ‘rule’ for dividing powers with like bases?”

• Allow students to talk or work together.
  • Hints you can give throughout:
    • Tell them to focus on the exponents in the original expression and the exponent in the simplified expression. How do those numbers relate?
    • Give them “a^m / aⁿ = a^?” and ask them to think about what happens with the original exponents to give us the final exponent.
• Eventually, talk through the quotient rule and explain how expanding each power allows us to cancel any common factors in both the numerator and denominator, and totally up the factors, we have left.
• Add the rule “a^m/aⁿ = a^m–n” to the classroom’s growing list of exponent rules.

Why teach the quotient rule of exponents this way?

Teaching the quotient rule of exponents this way allows students to discover the reasoning behind this rule. This creates a deeper conceptual understanding.

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Vocabulary for teaching the quotient rule of exponents

Exponent: An exponent (or power) is a small number placed to the upper-right of a base.

It shows how many times the base is multiplied by itself.

Power: The power of a number indicates how many times the base number will be multiplied.

Quotient Rule:  When multiplying powers with the same base, subtract their exponents.

a^m/aⁿ = a^m–n

Misconceptions and errors students are likely to have

• Students might divide the exponents instead of subtracting them.
  • Remind students of this activity. Encourage them to think about if they were to expand the powers, then cancel any common factors, how many would be leftover?
• Students may not consider the invisible exponent of 1 when a variable is written without an exponent
  • Remind them that any variable written without an exponent always has an invisible exponent of 1!

Resources:

• Teacher slides to use this activity in the classroom
• Downloadable worksheet for practicing the quotient rule of exponents
• Quiz for assessing the quotient rule of exponents

Frequently Asked Questions: