🔢 MathGrade 6Lesson 6
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Powers and Exponents (Intro)

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Learning Objectives

  • 1Identify the base and exponent in a power and read it aloud correctly
  • 2Write a power as repeated multiplication of its base
  • 3Evaluate small whole-number powers such as 2⁴, 5³, and 10²
  • 4Distinguish between a power like 3⁴ and a product like 3 × 4
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Mini Lesson

A power is a short way to write repeated multiplication. Instead of writing 5 × 5 × 5, we write 5³. The big number is the base — the number being multiplied. The small number floating up to the right is the exponent — it counts how many copies of the base are multiplied together.

Reading and Writing Powers

  • 5³ has a base of 5 and an exponent of 3.
  • It means "multiply 5 by itself 3 times": 5 × 5 × 5 = 125.
  • Read 5² as "five squared" and 5³ as "five cubed".
  • For other exponents, read 5⁴ as "five to the fourth power".
  • The exponent is the COUNT of bases, not a multiplier.

Power Form vs Expanded Form

  • Power form: 2⁵ — short and compact.
  • Expanded form: 2 × 2 × 2 × 2 × 2 — every factor is shown.
  • Both equal 32. The power form just hides the multiplications.
  • To turn expanded form into power form, count how many copies of the same base appear: 3 × 3 × 3 × 3 = 3⁴.

Special Cases — Exponents 0 and 1

  • Any non-zero base raised to the power 0 equals 1: 7⁰ = 1, 25⁰ = 1.
  • Any base raised to the power 1 equals the base itself: 7¹ = 7, 25¹ = 25.
  • These two rules are short-cuts, not multiplications you can act out.
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Common error: 3⁴ does NOT mean 3 × 4. 3⁴ = 3 × 3 × 3 × 3 = 81. 3 × 4 = 12. The exponent counts the bases — it is not a factor of multiplication.

Why Powers Matter

  • Powers of 10 (10¹, 10², 10³, …) name the place values: 10² = 100 (hundreds), 10³ = 1,000 (thousands).
  • Squares and cubes describe shapes: a 5×5 square has area 5² and a 5×5×5 cube has volume 5³.
  • Prime factorizations use exponents to write repeats: 24 = 2³ × 3.
Anatomy of a PowerThe power 5 cubed labeled with arrows pointing to the base 5 and the exponent 3, with the expanded form 5 times 5 times 5 equals 125.Anatomy of a Power53base = 5exponent = 35 x 5 x 5 = 125The exponent counts how many copies of the base are multiplied.
Base = what you multiply. Exponent = how many copies.
Common Powers TableA table of common powers of 2, 3, and 10 with their evaluated values from one up to the fifth power.Common Powers TablePowers of 2Powers of 3Powers of 102¹= 22²= 42³= 82⁴= 162⁵= 323¹= 33²= 93³= 2710¹= 1010²= 10010³= 1,000Any base to the 0 power equals 1. Anybase to the 1 power equals itself.
Memorize small powers of 2, 3, and 10 for fast work.
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Tip: when you evaluate a power, multiply ONE pair of bases at a time. For 2⁴: 2 × 2 = 4, then 4 × 2 = 8, then 8 × 2 = 16. Slow and steady beats trying to do all four at once.

Vocabulary
power
A number written using a base and an exponent, like 5³, that stands for repeated multiplication.
base
The number being multiplied in a power. In 5³, the base is 5.
exponent
The small raised number that tells how many copies of the base are multiplied. In 5³, the exponent is 3.
square
A power with exponent 2, like 5² — read "five squared". 5² = 25.
cube
A power with exponent 3, like 5³ — read "five cubed". 5³ = 125.
evaluate
To work out the value of a power by multiplying the bases — for example, evaluate 2⁴ as 16.
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