Powers, Indices, LCM and GCF
Primary 6 · Mathematics
Powers, Indices, LCM and GCF
Lesson 2 of 6
30 min
Let's begin
Ishimwe's teacher writes 2³ × 2⁴ on the board and asks for the answer as one power. Ishimwe writes out every 2 and counts them. “There must be a quicker way,” she says. There is!
2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2). Count the 2s: 3 + 4 = 7. So 2³ × 2⁴ = 2⁷.
Law 1: when you multiply powers of the same base, keep the base and add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ.
Worked example
10² × 10³ = 10²⁺³ = 10⁵ = 100 000. Check: 100 × 1 000 = 100 000. ✔
3⁵ ÷ 3² = (3 × 3 × 3 × 3 × 3) ÷ (3 × 3). Two 3s on the top cancel with the two 3s on the bottom. Three 3s are left: 3³.
Law 2: when you divide powers of the same base, keep the base and subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
Worked example
7⁸ ÷ 7⁵ = 7⁸⁻⁵ = 7³ = 343.
5³ ÷ 5³ = 5³⁻³ = 5⁰. But any number divided by itself is 1. So 5⁰ = 1. In fact, any non-zero number to the power 0 is 1.
2³ × 3² cannot be written as one power. Work it out instead: 8 × 9 = 72. It is not 6⁵.
Worked example
Simplify 2⁵ × 2³ ÷ 2⁴, then find its value.
The laws also work with letters: a³ × a⁴ = a⁷ and b⁶ ÷ b² = b⁴.
Tip
Times → add, divide → subtract. But only when the bases match, like counting mangoes with mangoes and not with avocados.
Work each one in your exercise book first, then tap Show answer to check.
Write 5⁴ × 5² as one power.
Answer: 5⁶ (4 + 2 = 6).
Write 9⁷ ÷ 9³ as one power.
Answer: 9⁴ (7 − 3 = 4).
Find the value of 3² × 3³ ÷ 3⁴.
Answer: 3¹ = 3. (2 + 3 − 4 = 1)
What is 12⁰?
Answer: 1. Any non-zero number to the power 0 is 1.
Simplify x⁵ × x.
Answer: x⁶. The x on its own is x¹.
What you learned today
Sources and credits
The stories, Rwandan worked examples and practice questions were written by Ganzaa.
Quick check
Write 3⁴ × 3⁵ as one power.
