Chapter 3 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

M4.1.3 Chapter 3

Roots and fractional powers

What can it mean to raise 9 to the power 1/2?

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

  • Find the cube root of 64 and of −8
  • Read 9 to the power 1/2 as √9
  • Work out 8 to the power 2/3

Kofi knows that √9 = 3, because 3 × 3 = 9. Then he meets 9 to the power 1/2. What can that be?

First, roots. The square root of a number is the positive number that gives it when multiplied by itself. The sign √ always means this positive root. Both 3 × 3 and (−3) × (−3) make 9, but √9 = 3.

Draw it first

A root is a side

  1. Step 1: a square of 9 tiles, 3 rows of 3, with 3 above and 3 on the left. Step 2: the label area 9. Step 3: the label side √9 = 3.33

    1 Lay 9 tiles in a square. It is 3 tiles along each side.

  2. Step 1: a square of 9 tiles, 3 rows of 3, with 3 above and 3 on the left. Step 2: the label area 9. Step 3: the label side √9 = 3.33area 9

    2 The area is the number of tiles: 3 × 3 = 9.

  3. Step 1: a square of 9 tiles, 3 rows of 3, with 3 above and 3 on the left. Step 2: the label area 9. Step 3: the label side √9 = 3.33area 9side √9 = 3

    3 The square root goes back from the area to the side: √9 = 3.

Now think of a power of 1/2. We choose a meaning for 9 to the power 1/2 that keeps the laws working. Call it r, so r = 9 to the power 1/2. Squaring is a power of a power, so the third law says to multiply the indices: 1/2 × 2 = 1. So r × r = 9¹ = 9.

Both 3 and −3 give 9 when squared. We take the positive one, so 9 to the power 1/2 = √9 = 3. In general, a to the power 1/2 is √a, for a not below 0.

The same idea works for a cube root. A power of 1/3 cubes back to the number, so a to the power 1/3 is the cube root of a.

A cube made of 64 small cubes, 4 along each edge, with 4 written beside three edges. The label under it says 4 × 4 × 4 = 64.4444 × 4 × 4 = 64
A cube of 64 small cubes has an edge of 4 cubes. So the cube root of 64 is 4.

Worked example

Find the cube root of 64.

  1. Look for a number that gives 64 when three copies of it are multiplied together.
  2. Try 3: 3 × 3 × 3 = 27. That is too small.
  3. Try 4: 4 × 4 × 4 = 64.
  4. So the cube root of 64 is 4.

Answer: The cube root of 64 is 4

A cube root can be of a negative number. Three copies of a negative number, multiplied together, give a negative number. The lines below show that the cube roots of −8 and 8 are on opposite sides of 0.

Two number lines from −4 to 4, one above the other. On the first, a point at −2 is labelled cube root of −8. On the second, a point at 2 is labelled cube root of 8.−4−3−2−101234cube root of −8−4−3−2−101234cube root of 8
The cube root of −8 is −2, and the cube root of 8 is 2.

Worked example

Find the cube root of −8.

  1. Start with 8. 2 × 2 × 2 = 8, so the cube root of 8 is 2.
  2. Try −2: (−2) × (−2) = 4, because two negatives make a positive. Then 4 × (−2) = −8.
  3. So the cube root of −8 is −2.

Answer: The cube root of −8 is −2

There is a limit. In the real numbers you cannot take an even root of a negative number. The real numbers are all the numbers on the number line. Any number multiplied by itself is 0 or more. So there is no square root of −9. An odd root, such as a cube root, of a negative number does exist.

Now a power with a top number. 2/3 is 1/3 × 2. So 8 to the power 2/3 is the cube root of 8, then squared. The bottom number is the root. The top number is the power.

On the left, a cube of 8 small cubes, 2 along each edge, labelled cube root: 2. On the right, a square of 4 tiles, 2 rows of 2, labelled squared: 4.cube root: 2squared: 4
8 to the power 2/3: first the cube root, 2. Then square it: 4.

Worked example

Work out 8 to the power 2/3.

  1. The bottom number 3 says: take the cube root first. The cube root of 8 is 2.
  2. The top number 2 says: square the result.
  3. 2² = 2 × 2 = 4.
  4. So 8 to the power 2/3 = 4.

Answer: 8 to the power 2/3 = 4

See it three ways

With things

A square floor has 16 tiles. How many tiles are along one side?

As a picture

A square of 16 tiles, 4 rows of 4, with 4 written above and 4 on the left.44

In symbols

16 to the power 1/2 = √16 = 4, because 4 × 4 = 16.

Try it

Hunt for a cube root

You need: paper and a pencil.

  1. Write the cubes of 1, 2, 3 and 4 in a list.
  2. Find 27 in your list.
  3. Say which number was cubed to make it.

What you will see: 3 × 3 × 3 = 27. So the cube root of 27 is 3. Reading the list backwards gives roots.

If you cannot try it, this is what happens.
A table with the head Number and Its cube. Rows: 1, 1; 2, 8; 3, 27; 4, 64.NumberIts cube1128327464

A wrong idea many people have

9 to the power 1/2 is half of 9, so it is 4.5.

Why it feels right: The index 1/2 looks like half, and half of 9 is 4.5.

The test that breaks it: Multiply 4.5 by itself: 4.5 × 4.5 = 20.25, not 9. But 3 × 3 = 9.

What is true: A power of 1/2 is a square root. 9 to the power 1/2 = √9 = 3.

A wrong idea many people have

The square root of −9 is −3.

Why it feels right: It seems that −3 is the number that goes with 9.

The test that breaks it: Check it: (−3) × (−3) = 9, not −9. Two negatives make a positive.

What is true: No real number multiplies by itself to give −9. An even root of a negative number does not exist in the real numbers. The cube root of −8 does exist, and it is −2.

Key words

square root
the positive number that gives a number when multiplied by itself. √9 = 3.
cube root
the number that gives a number when three copies of it are multiplied together. The cube root of −8 is −2.
real numbers
all the numbers on the number line, such as −2, 0, 1/2 and 3.

Back to your guess

The answer is B: 9 to the power 1/2 is the square root of 9, which is 3.

A power of 1/2 is a square root. Two copies of 9 to the power 1/2 multiply to 9¹, and 3 × 3 = 9. So the answer is 3. It is not half of 9 and not one over 9.

Check yourself

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

  1. The picture shows a square made of 25 tiles. What is 25 to the power 1/2?

    A square of 25 tiles, 5 rows of 5, with 5 written above and 5 on the left.55
    Tap your answer.

    1. Not this one. You took half of 25. A power of 1/2 is a square root, which is the side of the square, not half the area. Go to M4.1.3. Try another.

    2. Not this one. One over 25 is what a negative index gives. A positive index of 1/2 gives a root. Go to M4.1.2. Try another.

    3. Yes, that is right. The side of the square is 5 tiles, because 5 × 5 = 25. So 25 to the power 1/2 = √25 = 5.

    See the answer

    5

    The side of the square is 5 tiles, because 5 × 5 = 25. So 25 to the power 1/2 = √25 = 5.

    • If you chose A: You took half of 25. A power of 1/2 is a square root, which is the side of the square, not half the area. Go to M4.1.3.
    • If you chose B: One over 25 is what a negative index gives. A positive index of 1/2 gives a root. Go to M4.1.2.

  2. What is the cube root of −27?

    Tap your answer.

    1. Yes, that is right. (−3) × (−3) = 9, and 9 × (−3) = −27. So the cube root of −27 is −3.

    2. Not this one. You dropped the minus sign. 3 × 3 × 3 = 27, not −27. A negative number cubed stays negative. Go to M4.1.3. Try another.

    3. Not this one. Only even roots of negative numbers fail. A cube root of a negative number exists, because three negatives multiply to a negative. Go to M4.1.3. Try another.

    4. Not this one. You divided by 3. A cube root is the number whose three copies, multiplied together, give the number. Go to M4.1.3. Try another.

    See the answer

    −3

    (−3) × (−3) = 9, and 9 × (−3) = −27. So the cube root of −27 is −3.

    • If you chose B: You dropped the minus sign. 3 × 3 × 3 = 27, not −27. A negative number cubed stays negative. Go to M4.1.3.
    • If you chose C: Only even roots of negative numbers fail. A cube root of a negative number exists, because three negatives multiply to a negative. Go to M4.1.3.
    • If you chose D: You divided by 3. A cube root is the number whose three copies, multiplied together, give the number. Go to M4.1.3.

  3. Why is there no square root of −16 among the real numbers?

    Tap your answer.

    1. Not this one. A negative times a negative is positive: −4 × −4 = +16. So −4 does not square to −16. Go to M3.1.2. Try another.

    2. Yes, that is right. Positive times positive is positive, and negative times negative is positive. Zero times zero is zero. So a square is never below 0.

    3. Not this one. Cube roots of negative numbers exist: the cube root of −8 is −2. Only even roots, such as square roots, fail. Go to M4.1.3. Try another.

    See the answer

    The square of a real number is never negative, so no real number squares to give it.

    Positive times positive is positive, and negative times negative is positive. Zero times zero is zero. So a square is never below 0.

    • If you chose A: A negative times a negative is positive: −4 × −4 = +16. So −4 does not square to −16. Go to M3.1.2.
    • If you chose C: Cube roots of negative numbers exist: the cube root of −8 is −2. Only even roots, such as square roots, fail. Go to M4.1.3.

  4. Work out 64 to the power 2/3.

    Tap your answer.

    1. Not this one. You took the cube root, which is 4, and stopped. The top number 2 still says to square the result. Go to M4.1.3. Try another.

    2. Not this one. You multiplied 64 by 2/3. A power is not a multiplier. The bottom number is a root and the top number is a power. Go to M4.1.3. Try another.

    3. Not this one. You squared 64 and ignored the bottom number. The bottom number 3 says to take a cube root as well. Go to M4.1.3. Try another.

    4. Yes, that is right. The cube root of 64 is 4. Squared, that is 4 × 4 = 16.

    See the answer

    16

    The cube root of 64 is 4. Squared, that is 4 × 4 = 16.

    • If you chose A: You took the cube root, which is 4, and stopped. The top number 2 still says to square the result. Go to M4.1.3.
    • If you chose B: You multiplied 64 by 2/3. A power is not a multiplier. The bottom number is a root and the top number is a power. Go to M4.1.3.
    • If you chose C: You squared 64 and ignored the bottom number. The bottom number 3 says to take a cube root as well. Go to M4.1.3.

  5. The block is a cube of 125 small cubes. What is the cube root of 125?

    A cube made of 125 small cubes, 5 along each edge, with 5 written beside three edges.555
    Tap your answer.

    1. Yes, that is right. 5 × 5 × 5 = 125, so the edge is 5 small cubes long.

    2. Not this one. 25 is the number of small cubes in one layer, 5 × 5. The cube root is the edge, the number you multiply three times: 5 × 5 × 5. Go to M4.1.3. Try another.

    3. Not this one. You divided 125 by 3. The cube root is the edge of the cube, the number you multiply three times. Go to M4.1.3. Try another.

    See the answer

    5

    5 × 5 × 5 = 125, so the edge is 5 small cubes long.

    • If you chose B: 25 is the number of small cubes in one layer, 5 × 5. The cube root is the edge, the number you multiply three times: 5 × 5 × 5. Go to M4.1.3.
    • If you chose C: You divided 125 by 3. The cube root is the edge of the cube, the number you multiply three times. Go to M4.1.3.

  6. From an earlier chapter

    Write 4⁻² as a fraction with 1 on top, such as 1/8.

    See the answer

    1/16

    4⁻² = 1/4², and 4² = 16.

  7. What is the cube root of 216?

    See the answer

    6

    6 × 6 × 6 = 216, so the cube root of 216 is 6.

  8. Work out 16 to the power 1/2 + 27 to the power 1/3.

    See the answer

    7

    16 to the power 1/2 = 4, because 4 × 4 = 16. 27 to the power 1/3 = 3, because 3 × 3 × 3 = 27. Together: 4 + 3 = 7.

  9. A cube has a volume of 343 cm³. How long is each edge, in cm?

    See the answer

    7

    The edge is the number that gives 343 when multiplied three times: 7 × 7 × 7 = 343. So each edge is 7 cm.

Keep it

Copy this into your exercise book.

A power of 1/2 is a square root. A power of 1/3 is a cube root. With 2/3: root first, then power. The cube root of −8 is −2. No real square root of −9.

Draw: A square of 9 tiles with side 3, and next to it a cube of 8 small cubes with edge 2.

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

No internet tomorrow? Save the whole book as one file (755 KB). It opens on any phone.