Chapter 1 of 8

All 8 chapters
  1. 1Laws of powers, readunderstood
  2. 2Zero and negative powers, readunderstood
  3. 3Roots and fractional powers, readunderstood
  4. 4Standard form, readunderstood
  5. 5Calculating in standard form, readunderstood
  6. 6Rational and irrational numbers, readunderstood
  7. 7Surds, readunderstood
  8. 8Accuracy and bounds, readunderstood

M4.1.1 Chapter 1

Laws of powers

Why is a³ × a⁴ equal to a⁷, and not a¹²?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Simplify a³ × a⁴ and a⁷ ÷ a²
  • Simplify (a³)² and (2a²)³

Ravi sees a³ × a⁴ on a page. He asks how many a's are being multiplied together. Here a stands for any number.

In a³, the letter a is the base. The small raised 3 is the index. It counts the copies of a that are multiplied. So a³ = a × a × a. The plural of index is indices.

Draw it first

Put the a's in boxes

  1. Step 1: three boxes marked a with the count 3 under them, under the label a³ × a⁴. Step 2: the same three boxes, then a times sign, then four boxes marked a with the count 4. Step 3: a second row of seven boxes marked a with the count 7, and the label = a⁷.a³ × a⁴aaa3

    1 a³ means three a's multiplied together. Draw them as three boxes.

  2. Step 1: three boxes marked a with the count 3 under them, under the label a³ × a⁴. Step 2: the same three boxes, then a times sign, then four boxes marked a with the count 4. Step 3: a second row of seven boxes marked a with the count 7, and the label = a⁷.a³ × a⁴aaa3×aaaa4

    2 a⁴ means four a's. Draw four more boxes after the × sign.

  3. Step 1: three boxes marked a with the count 3 under them, under the label a³ × a⁴. Step 2: the same three boxes, then a times sign, then four boxes marked a with the count 4. Step 3: a second row of seven boxes marked a with the count 7, and the label = a⁷.a³ × a⁴aaa3×aaaa4aaaaaaa7= a⁷

    3 Now all the boxes are multiplied. Count them: 3 + 4 = 7 boxes, so the product is a⁷.

This is the first law. To multiply powers of the same base, add the indices. It only works when the base is the same.

Worked example

Simplify a³ × a⁴.

  1. a³ is 3 a's multiplied together.
  2. a⁴ is 4 a's. Together there are 3 + 4 = 7 a's.
  3. Seven a's multiplied together is a⁷.

Answer: a³ × a⁴ = a⁷

Now divide. a⁷ ÷ a² is a fraction with seven a's above and two a's below. A letter divided by itself is 1, because a ÷ a = 1 whenever a is not 0. So each a below cancels one a above.

A row of 7 boxes marked a over a row of 2 boxes marked a, with a fraction bar between them. The first 2 boxes in the top row and both boxes in the bottom row are crossed out in pairs. 5 left is written under the top row.aaaaaaaaa725 left
a⁷ ÷ a²: two pairs cross out, and 5 a's are left.

This is the second law. To divide powers of the same base, subtract the indices: the index on top minus the index below.

Worked example

Simplify a⁷ ÷ a².

  1. Write it as a fraction. There are 7 a's above and 2 a's below.
  2. Each a below cancels one a above, because a ÷ a = 1.
  3. 2 pairs cancel, leaving 7 − 2 = 5 a's above.
  4. So the answer is a⁵.

Answer: a⁷ ÷ a² = a⁵

A power can be raised to a power. (a³)² means two copies of a³ multiplied together.

Two groups of 3 boxes marked a, with a times sign between them. Each group has the count 3 under it. The label above says (a³)² = a³ × a³.(a³)² = a³ × a³aaa3×aaa3
(a³)² is 2 groups of 3 a's: 2 × 3 = 6 a's.

This is the third law. To raise a power to a power, multiply the indices. When the inside of the bracket is a product, such as 2 × a², the index applies to every part of it. This does not work for a sum: (a + b)² is not a² + b².

Worked example

Simplify (2a²)³.

  1. The index 3 means 3 copies are multiplied: 2a² × 2a² × 2a².
  2. Numbers first: 2 × 2 × 2 = 8.
  3. Letters next: a² × a² × a² has 2 + 2 + 2 = 6 a's multiplied.
  4. Together, (2a²)³ = 8a⁶.

Answer: (2a²)³ = 8a⁶

See it three ways

With things

Write a in a box 2 times, then 3 more times. Count the boxes.

As a picture

Two boxes marked a with the count 2 under them, a times sign, then three boxes marked a with the count 3 under them.aa2×aaa3

In symbols

a² × a³ = a × a × a × a × a = a⁵, and 2 + 3 = 5.

A table of the three laws with the head Indices and Example. Rows: add, a³ × a⁴ = a⁷; subtract, a⁷ ÷ a² = a⁵; multiply, (a³)² = a⁶.IndicesExampleadda³ × a⁴ = a⁷subtracta⁷ ÷ a² = a⁵multiply(a³)² = a⁶
What to do with the indices: add them to multiply, subtract them to divide, multiply them for a power of a power.

Try it

Test the first law with numbers

You need: paper and a pencil.

  1. Work out 2³ = 2 × 2 × 2 and 2⁴ = 2 × 2 × 2 × 2.
  2. Multiply the two answers together.
  3. Now work out 2⁷ and compare.

What you will see: 2³ = 8 and 2⁴ = 16. Then 8 × 16 = 128, and 2⁷ = 128 as well. The law gives the same answer as working it out.

If you cannot try it, this is what happens.
A table with the head Sum and Value. The row 2³ × 2⁴ = 8 × 16 has the value 128. The row 2⁷ has the value 128.SumValue2³ × 2⁴ = 8 × 161282⁷128

A wrong idea many people have

a³ × a⁴ = a¹², because you multiply the indices.

Why it feels right: The sign says times, so it feels as if everything should be multiplied.

The test that breaks it: Put a = 2. Then 2³ × 2⁴ = 8 × 16 = 128, but 2¹² = 4096. The two do not match, and 2⁷ = 128 does.

What is true: When you multiply powers of the same base, you add the indices. You multiply indices only when a power is raised to a power.

A wrong idea many people have

a³ × b⁴ = ab⁷.

Why it feels right: The indices 3 and 4 are there to be added.

The test that breaks it: Put a = 2 and b = 3. Then a³ × b⁴ = 8 × 81 = 648, but ab⁷ = 2 × 2187 = 4374.

What is true: The laws need the same base. a³ × b⁴ has two different letters, so it stays as it is.

Key words

base
the number or letter whose copies are multiplied, such as a in a³.
index
the small raised number. It counts the copies of the base that are multiplied.
indices
more than one index.

Back to your guess

The answer is C: a³ × a⁴ = a⁷, because there are 3 + 4 = 7 letters a multiplied together.

Three a's multiplied by four a's make seven a's multiplied together. So the answer is a⁷. You add the indices. You do not multiply them.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Simplify a⁵ × a².

    Tap your answer.

    1. Not this one. You multiplied the indices. When the same base is multiplied, the a's just join up, so the indices add. Go to M4.1.1. Try another.

    2. Yes, that is right. 5 a's and 2 more a's make 5 + 2 = 7 a's.

    3. Not this one. You subtracted the indices. Subtracting is for division. A product has more a's than either part. Go to M4.1.1. Try another.

    See the answer

    a⁷

    5 a's and 2 more a's make 5 + 2 = 7 a's.

    • If you chose A: You multiplied the indices. When the same base is multiplied, the a's just join up, so the indices add. Go to M4.1.1.
    • If you chose C: You subtracted the indices. Subtracting is for division. A product has more a's than either part. Go to M4.1.1.

  2. The picture shows a⁶ ÷ a². Which answer is right?

    A row of 6 boxes marked a over a row of 2 boxes marked a, with a fraction bar between them. Pairs are crossed out. 4 left is written under the top row.aaaaaaaa624 left
    Tap your answer.

    1. Not this one. You divided the indices. Dividing powers takes away a's, so the indices are subtracted. Go to M4.1.1. Try another.

    2. Not this one. You added the indices. Adding is for multiplying. Here the a's below cancel some of the a's above. Go to M4.1.1. Try another.

    3. Yes, that is right. 2 pairs cross out. 6 − 2 = 4 a's are left.

    See the answer

    a⁴

    2 pairs cross out. 6 − 2 = 4 a's are left.

    • If you chose A: You divided the indices. Dividing powers takes away a's, so the indices are subtracted. Go to M4.1.1.
    • If you chose B: You added the indices. Adding is for multiplying. Here the a's below cancel some of the a's above. Go to M4.1.1.

  3. Simplify (a³)².

    Tap your answer.

    1. Yes, that is right. 2 groups of 3 a's make 2 × 3 = 6 a's.

    2. Not this one. You added the indices. Here a³ is written twice and the two copies are multiplied: that is 2 groups of 3 a's. Go to M4.1.1. Try another.

    3. Not this one. You worked out 3 squared. The 2 outside the bracket counts the copies of a³. It does not square the 3. Go to M4.1.1. Try another.

    See the answer

    a⁶

    2 groups of 3 a's make 2 × 3 = 6 a's.

    • If you chose B: You added the indices. Here a³ is written twice and the two copies are multiplied: that is 2 groups of 3 a's. Go to M4.1.1.
    • If you chose C: You worked out 3 squared. The 2 outside the bracket counts the copies of a³. It does not square the 3. Go to M4.1.1.

  4. Simplify (3a⁴)².

    Tap your answer.

    1. Not this one. You squared the a⁴ but not the 3. The bracket squares everything inside it, so the 3 is squared as well. Go to M4.1.1. Try another.

    2. Not this one. You multiplied 3 by 2. Squaring 3 means 3 × 3, which is 9. Go to M4.1.1. Try another.

    3. Not this one. You added 4 and 2 for the index. The index of a⁴ is multiplied by the 2 outside: 4 × 2. Go to M4.1.1. Try another.

    4. Yes, that is right. Square the number: 3 × 3 = 9. Multiply the indices: 4 × 2 = 8.

    See the answer

    9a⁸

    Square the number: 3 × 3 = 9. Multiply the indices: 4 × 2 = 8.

    • If you chose A: You squared the a⁴ but not the 3. The bracket squares everything inside it, so the 3 is squared as well. Go to M4.1.1.
    • If you chose B: You multiplied 3 by 2. Squaring 3 means 3 × 3, which is 9. Go to M4.1.1.
    • If you chose C: You added 4 and 2 for the index. The index of a⁴ is multiplied by the 2 outside: 4 × 2. Go to M4.1.1.

  5. Can a³ × b⁴ be written as one power of a letter by adding the indices?

    Tap your answer.

    1. Not this one. The indices only join up when the base is the same. A power of a and a power of b cannot be combined this way. Go to M4.1.1. Try another.

    2. Not this one. Adding or multiplying indices needs one base. With two different bases the product stays as it is. Go to M4.1.1. Try another.

    3. Yes, that is right. The first law needs the same base. a³ × b⁴ has two bases, so it cannot be simplified.

    See the answer

    No. The letters are different, so the indices cannot be added.

    The first law needs the same base. a³ × b⁴ has two bases, so it cannot be simplified.

    • If you chose A: The indices only join up when the base is the same. A power of a and a power of b cannot be combined this way. Go to M4.1.1.
    • If you chose B: Adding or multiplying indices needs one base. With two different bases the product stays as it is. Go to M4.1.1.

  6. Simplify a⁸ ÷ a³. What is the index in the answer?

    See the answer

    5

    Subtract the indices: 8 − 3 = 5. So a⁸ ÷ a³ = a⁵.

  7. Simplify a⁴ × a⁵ ÷ a³. What is the index in the answer?

    See the answer

    6

    Multiply first: a⁴ × a⁵ = a⁹. Then divide: 9 − 3 = 6.

  8. Work out 2⁶ ÷ 2⁴ as a power of 2, and then as a number. What is the number?

    See the answer

    4

    Subtract the indices: 6 − 4 = 2. So 2⁶ ÷ 2⁴ = 2² = 4. Check: 64 ÷ 16 = 4.

  9. Simplify (a⁴)³ × a². What is the index in the answer?

    See the answer

    14

    First the bracket: 3 × 4 = 12, so (a⁴)³ = a¹². Then add: 12 + 2 = 14.

Keep it

Copy this into your exercise book.

Same base: multiply, add the indices. Divide, subtract them. Power of a power, multiply them. A bracket powers the number too: (2a²)³ = 8a⁶.

Draw: Seven boxes marked a, split into 3 and 4, with a³ × a⁴ = a⁷ written under them.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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