Exponents & Scientific Notation — Print

An exponent counts how many times you multiply, not what you multiply by. That one idea explains every rule.

What the practice questions cover

These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Fold a sheet of paper in half. Now fold it again, and again.

One fold gives 2 layers. Two folds give 4. Three give 8.

fold a sheet of paper in half, again and again 0 folds 1 1 fold 2 2 folds 4 3 folds 8 4 folds 16 seven folds and it is already 128 sheets thick writing 2 × 2 × 2 × 2 × 2 × 2 × 2 is slow, and easy to miscount

Each fold doubles what is already there, so by seven folds you are at 128 layers.

Writing that as 2 × 2 × 2 × 2 × 2 × 2 × 2 is slow, and easy to miscount.

So write 27 instead. The small raised number simply counts the twos.

That is all an exponent is: a count of how many copies are multiplied. Every rule in this unit — adding exponents, subtracting them, the meaning of 20 — can be rebuilt by asking how many copies there are.

Doubling: each step multiplies by two 22^142^282^3162^4322^5 an exponent counts factors — each step multiplies by 2, it does not add 2 2^5 = 32, not 10

What an exponent actually means

53 means 5 × 5 × 5 = 125. It does not mean 5 × 3.

Read it as "five to the third power", or "five cubed". a positive whole-number exponent counts factors. Zero, negative and fractional exponents extend that idea.

Once you hold on to that, you never have to memorise the rules — you can rebuild them.

The most common slip

34 ≠ 12. It is 3 × 3 × 3 × 3 = 81. Multiplying the base by the exponent is the single most frequent error in this unit.

Multiplying and dividing powers

23 × 24 = (2 · 2 · 2)(2 · 2 · 2 · 2) = 27. Three factors and four factors make seven factors. So you add the exponents.

2724 cancels four of the seven twos, leaving 23. So you subtract.

Both rules only work when the bases match. 23 × 34 cannot be combined at all.

The three rules

am · an = am+n
aman = am-n
(am)n = amn — a power of a power multiplies.

Zero and negative exponents

Follow the pattern down: 23 = 8, 22 = 4, 21 = 2. Each step divides by 2. The next step gives 20 = 1.

Keep going: 2-1 = 12 and 2-2 = 14.

So a negative exponent means "one over", not "negative number". 2-3 = 18, which is positive.

Say it out loud

A negative exponent flips the number to the other side of the fraction bar. It does not change the sign by itself; a negative base can still give a negative result, such as (-2)-3 = -1/8.

Scientific notation

Very large and very small numbers are painful to write out. Scientific notation packs them into a × 10n with exactly one non-zero digit in front of the decimal point.

4,300,000 = 4.3 × 106. The point moved 6 places left, so the exponent is 6.

0.00047 = 4.7 × 10-4. The point moved 4 places right, so the exponent is -4.

Big number, positive exponent. Small number, negative exponent. Check that first, every time.

Calculating in scientific notation

Multiply the front numbers, and add the exponents: (3 × 105)(2 × 10-2) = 6 × 103.

If the front number lands outside 1 to 10, fix it. (5 × 104)(4 × 103) = 20 × 107, which becomes 2 × 108.

Worked examples

Combining rules

Simplify (23)2 · 2425.

  1. (23)2 = 26 A power of a power multiplies: 3 × 2 = 6.
  2. Top becomes 26 · 24 = 210 Same base multiplied, so add the exponents.
  3. 21025 = 25 Dividing subtracts: 10 - 5 = 5.
  4. 25 = 32 Five twos multiplied together.

25 = 32

Into scientific notation

Write 0.00062 in scientific notation.

  1. Put one non-zero digit before the point: 6.2 That is the a part, between 1 and 10.
  2. Count the moves: 0.00062 → 6.2 is 4 places right. Moving right means the original was small.
  3. Small number, so the exponent is negative: 10-4 Check: 6.2 × 0.0001 = 0.00062 ✓

6.2 × 10-4

Multiplying big numbers

Compute (4 × 106)(5 × 10-2) in scientific notation.

  1. Front numbers: 4 × 5 = 20 Multiply the a parts.
  2. Powers: 106 · 10-2 = 104 Add the exponents: 6 + (-2) = 4.
  3. So far 20 × 104 — but 20 is too big. The front must sit between 1 and 10.
  4. 20 = 2 × 101, so the answer is 2 × 105 One more factor of ten moves across.

2 × 105

Words to know

Base
The number being multiplied. In 53 the base is 5.
Exponent
How many times the base is used as a factor. In 53 the exponent is 3.
Power
The whole expression, 53, and also its value, 125.
Scientific notation
A number written as a × 10n, where 1 ≤ a < 10.

Mistakes to watch for

Multiplying the base by the exponent. 34 is 3 × 3 × 3 × 3 = 81, not 12 — the exponent counts the factors, it is not one of them. This is the single most frequent error in the unit.

Thinking 2-3 is negative. It is 18: a negative exponent means repeated division, not a negative result. Only a minus in front of the whole thing makes it negative.

Adding exponents with different bases. 23 · 32 cannot be combined, because they are counts of different things — three twos and two threes. The rule only works when the bases match.

Leaving a front number outside 1 to 10 in scientific notation. 47 × 103 has the right value but breaks the form, and two numbers written that way can no longer be compared by exponent.

The word exponent comes from Latin exponere, "to put out" — it is the number put out above the base.

Before the modern notation, mathematicians wrote x · x · x in full. The superscript form was popularised by René Descartes in the 1630s.

There is a long-standing claim that paper cannot be folded more than seven times. In 2002 a student named Britney Gallivan folded one twelve times, using 1.2 km of toilet paper and a formula she derived herself.

Problem 1

Evaluate 34.

Worked answer

81

3 × 3 × 3 × 3 = 81

Problem 2

Evaluate 25.

Worked answer

32

2 × 2 × 2 × 2 × 2 = 32

Problem 3

Evaluate 70.

Worked answer

1

Follow the pattern down: 72 = 49, 71 = 7, 70 = 1.

Problem 4

Simplify 52 · 53 and give its value.

Worked answer

3125

52+3 = 55

= 3125

Problem 5

Write 4.3 × 106 in standard form.

Worked answer

4300000

4.3 × 1,000,000 = 4,300,000

Problem 6

Simplify 23 · 24 and give its value.

Worked answer

128

23+4 = 27

= 128

Problem 7

Simplify 3835 and give its value.

Worked answer

27

38-5 = 33

= 27

Problem 8

Simplify (23)2 and give its value.

Worked answer

64

23 × 2 = 26

= 64

Problem 9

Evaluate 2-3.

Worked answer

1/8

2-3 = 123

= 18

Problem 10

Write 0.00047 in scientific notation. Enter just the exponent of 10.

Worked answer

-4

0.00047 → 4.7 is 4 places right

Small number, so negative: 4.7 × 10-4

Problem 11

Write 52,000 in scientific notation. Enter just the exponent of 10.

Worked answer

4

52,000 → 5.2 is 4 places left

5.2 × 104

Problem 12

Compute (3 × 105)(2 × 10-2) and give the answer in standard form.

Worked answer

6000

3 × 2 = 6

105 · 10-2 = 103

6 × 103 = 6000

Problem 13

Simplify (23)2 · 2425 and give its value.

Worked answer

32

(23)2 = 26

Top: 26 · 24 = 210

21025 = 25 = 32

Problem 14

Compute (5 × 104)(4 × 103) in scientific notation. Enter just the exponent of 10.

Worked answer

8

5 × 4 = 20 and 104 · 103 = 107

20 × 107 = 2 × 101 × 107

= 2 × 108

Problem 15

Which is larger, 210 or 103?

  1. 210
  2. 103
  3. They are equal
Worked answer

210

210 = 1024

103 = 1000

1024 > 1000

Problem 16

Evaluate (-2)4 + (-2)3.

Worked answer

8

(-2)4 = 16 — four negatives, even

(-2)3 = -8 — three negatives, odd

16 + (-8) = 8

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