Math Vale

🔢 Number Kingdom · Number Theory

Exponents

Use exponents as shorthand for repeated multiplication, and apply the product, quotient, zero and negative exponent rules.

In short

  • An exponent counts factors: 34 is 3 x 3 x 3 x 3, not 3 x 4.
  • Same base: multiplying adds the exponents, dividing subtracts them. The base itself never changes.
  • Any non-zero number to the power 0 is 1, and a negative exponent means a reciprocal.
  • An exponent binds only to the base directly beneath it, and is evaluated before multiplication and addition.

Shorthand for repeated multiplication

34 means 3 x 3 x 3 x 3 = 81. The base is 3 and the exponent is 4; the exponent counts how many copies of the base are multiplied together.

The single most common error in this whole topic is reading 34 as 3 x 4 = 12. The exponent is not a factor — it is a count of factors.

Two names worth knowing: 52 is read "5 squared" (the area of a square with side 5), and 53 is "5 cubed" (the volume of a cube with side 5).

The product and quotient rules

Write the powers out and the rules become obvious rather than magic.

23 x 24 = (2 x 2 x 2) x (2 x 2 x 2 x 2) — that is 7 copies of 2, so it is 27.

  • Product rule: with the same base, multiplying ADDS the exponents. am x an = am+n.

25 / 22 puts 5 copies of 2 over 2 copies, and each one underneath cancels one above, leaving 3.

  • Quotient rule: with the same base, dividing SUBTRACTS the exponents. am / an = am-n.

Both rules need the same base. 23 x 52 cannot be combined this way, and the base never changes when you apply them: 23 x 24 is 27, never 47.

Zero and negative exponents

Follow the pattern downwards, dividing by the base at each step:

24 = 16 23 = 8 22 = 4 21 = 2 20 = 1 2-1 = 1/2 2-2 = 1/4

Each line is the one above divided by 2, and the pattern does not stop at zero. So:

  • a0 = 1 for any non-zero a. It is not 0, and it is not a.
  • a-n = 1 / an. A negative exponent means a reciprocal, not a negative answer. 5-2 = 1/25, which is positive.

The quotient rule says the same thing: 23 / 23 = 20, and any non-zero number divided by itself is 1.

Powers inside expressions

Exponents are evaluated before multiplication, division, addition and subtraction, and an exponent attaches only to the base immediately beneath it.

  • 3 x 42 = 3 x 16 = 48. The 3 is not squared.
  • (3 x 4)2 = 122 = 144. Brackets change what is being squared.
  • -52 = -(52) = -25, while (-5)2 = 25.

If you want the whole thing squared, you must write the brackets.

Why exponents matter

Powers grow ferociously fast, which is exactly why they are useful.

Doubling every day starting from 1 gives 1, 2, 4, 8, 16, ... and after 30 days you pass a billion. That is 230.

Powers of ten are the backbone of place value (4,738 = 4 x 103 + 7 x 102 + 3 x 101 + 8 x 100), and they reappear in area and volume, in the Pythagorean theorem, and in every exponential growth model you will meet later.

Worked examples

Example 1

Write 56 / 54 as a single power, then evaluate it.

  1. The bases match, so use the quotient rule: subtract the exponents.
  2. 56 / 54 = 56-4 = 52.
  3. 52 = 5 x 5 = 25.
  4. Sanity check by expanding: 15,625 / 625 = 25.

Example 2

Evaluate 4 x 33 + 7.

  1. Exponents come first, and the exponent applies only to the 3.
  2. 33 = 3 x 3 x 3 = 27.
  3. Then multiplication: 4 x 27 = 108.
  4. Addition last: 108 + 7 = 115.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

42 = ?

Answer: 16

  1. 42 = 4 x 4
  2. = 16

Problem 2

Difficulty 3 of 5

82 x 87 can be written as 8k for a single exponent k. What is k?

Answer: 9

  1. 82 x 87 = (8 x 8) x (8 x 8 x 8 x 8 x ...)
  2. That is 2 + 7 = 9 copies of 8.
  3. So k = 9.

Problem 3

Difficulty 4 of 5

78 / 75 can be written as 7k for a single exponent k. What is k?

Answer: 3

  1. 78 / 75: 8 copies of 7 divided by 5 copies.
  2. Cancelling 5 pairs leaves 8 - 5 = 3 copies.
  3. So k = 3.

Common mistakes

  • Multiplying the base by the exponent: reading 34 as 12.
  • Multiplying the exponents when the powers are multiplied, or multiplying the bases as well.
  • Claiming a0 = 0, or that a negative exponent makes the answer negative.
  • Applying an exponent to a coefficient: reading 3 x 42 as 144.

What you should be able to do

  • Evaluate a power such as 34 and explain what the exponent counts.
  • Apply the product and quotient rules for powers with the same base.
  • Interpret a zero exponent and a negative exponent.
  • Evaluate powers inside expressions with the correct order of operations.

Where this fits in the curriculum

Common Core

  • 6.EE.A.1

    Grade 6 — Write and evaluate numerical expressions involving whole-number exponents.

  • 8.EE.A.1

    Grade 8 — Know and apply the properties of integer exponents, including zero and negative exponents.

Ontario

  • G7.B2.7

    Grade 7 — Evaluate and express repeated multiplication of whole numbers using exponential notation.

  • MTH1W.B2.2

    Grade 9 de-streamed — Apply the exponent laws to powers with the same base, and to a power of a power.

SAT

Learn these first

This leads on to