Year 9 maths lesson · Indices and Square Roots

Expanded Form and Index Form

This lesson teaches you how to convert between expanded form (e.g. 5 × 5 × 5) and index form (e.g. 5³). You’ll learn what the base and the power (index) mean, how to evaluate simple powers, and how to simplify products of powers that share the same base.

  • About 2–5 minutes
  • 3 quick questions as you go
  • Then about 4 minutes of practice
Practice questions → Set for Class

Worked examples

Evaluate 2⁴

  1. Expand: 2 × 2 × 2 × 2
  2. Calculate: 2 × 2 = 4, × 2 = 8, × 2 = 16
  3. Answer: 2⁴ = 16

Write 7 × 7 × 7 × 7 × 7 in index form

  1. Count: five 7s multiplying together
  2. Base is 7, power is 5
  3. Answer: 7⁵

Simplify 5⁴ × 5²

  1. Expand: (5 × 5 × 5 × 5) × (5 × 5)
  2. Count: four 5s + two 5s = six 5s
  3. Answer: 5⁶

Simplify 7⁴ × 7³ × 7²

  1. Expand each: (7×7×7×7) × (7×7×7) × (7×7)
  2. Count: 4 + 3 + 2 = nine 7s
  3. Answer: 7⁹

Before this lesson

It helps to be comfortable with:

  • Multiplication of whole numbers
  • Repeated multiplication concept

Lesson transcript

Everything the lesson says, word for word

When we multiply a number by itself over and over, writing it all out gets tedious. Index form gives us a shortcut. Instead of writing 5 times 5 times 5, we write 5 to the power of 3. The 5 is the base. The 3 is the power — it tells us how many times the base is multiplied.

Two to the power of four. That’s four 2s multiplying: 2 times 2 times 2 times 2, which equals 16.

Seven times seven times seven times seven times seven. That’s five 7s multiplying — so we write 7 to the power of 5.

5 to the 4, times 5 squared. Expand them: four 5s, times two 5s. Altogether that’s six 5s multiplying — 5 to the 6.

7 to the 4, times 7 cubed, times 7 squared. Expand each one: four 7s, three 7s, two 7s. Altogether that’s nine 7s multiplying — 7 to the 9.

More Indices and Square Roots lessons

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