Chapter 7 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

M4.1.7 Chapter 7

Surds

Why can √50 be written as 5√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

  • Simplify √50 to 5√2
  • Add and multiply surds, like √8 + 3√2
  • Rationalise the denominator of 6/√3

Yusuf finds that a square has an area of 50 cm². He wants its side exactly. The side is √50 cm, and a calculator only gives a rounded answer.

A surd is an exact root that is not a whole number or fraction, such as √2 or √50. It is left as a root. The decimal 1.414... is only an approximation of √2. And √9 is not a surd, because it is 3.

A square factor of a number is a factor that is a square number, such as 25 for 50. To simplify a surd, find the largest square factor, split the root, and take the root of the square part.

To simplify a surd we use one rule: for positive numbers, √a × √b = √(a × b). For example √4 × √9 = 2 × 3 = 6, and √(4 × 9) = √36 = 6. Both sides agree.

Here is why. Square the left side: (√a × √b)² = √a × √a × √b × √b = a × b. So √a × √b is the positive number whose square is a × b. That number is √(a × b).

A table with the head Sum and Value. Rows: √4 × √9, 2 × 3 = 6; √(4 × 9), √36 = 6.SumValue√4 × √92 × 3 = 6√(4 × 9)√36 = 6
Splitting a product under a root gives the same number.

Draw it first

Pull a pair out of the root

  1. A factor tree. Step 1: 50 at the top. Step 2: 50 splits into 25 and 2. Step 3: 25 splits into 5 and 5, so the leaves are 5, 5 and 2. Step 4: the label √50 = 5√2.50

    1 Start with 50 under the root sign.

  2. A factor tree. Step 1: 50 at the top. Step 2: 50 splits into 25 and 2. Step 3: 25 splits into 5 and 5, so the leaves are 5, 5 and 2. Step 4: the label √50 = 5√2.50252

    2 Split 50 into 25 × 2. Now 25 is a square number.

  3. A factor tree. Step 1: 50 at the top. Step 2: 50 splits into 25 and 2. Step 3: 25 splits into 5 and 5, so the leaves are 5, 5 and 2. Step 4: the label √50 = 5√2.5025552

    3 Split 25 into 5 × 5. A pair of 5s comes out of the root as one 5.

  4. A factor tree. Step 1: 50 at the top. Step 2: 50 splits into 25 and 2. Step 3: 25 splits into 5 and 5, so the leaves are 5, 5 and 2. Step 4: the label √50 = 5√2.5025552√50 = 5√2

    4 Write what is left: √50 = 5√2.

Worked example

Simplify √50.

  1. The largest square number that divides 50 is 25, because 50 = 25 × 2.
  2. Split the root: √50 = √25 × √2.
  3. √25 = 5.
  4. So √50 = 5√2.

Answer: √50 = 5√2

Here is a picture of why. A square of area 50 cm² can be cut into 25 tiny squares, each with area 2 cm². Each tiny square has a side of √2 cm. There are 5 of them along each side.

A square made of 25 tiles, 5 across and 5 down. The label above says 5√2 cm. The labels below say area 50 cm² and each tile 2 cm².5√2 cmarea 50 cm²each tile 2 cm²
Each tile has area 2 cm², so its side is √2 cm. Five tiles make a side of 5√2 cm.
A table with the head Surd, Factor and Simplified. Rows: √8, 4, 2√2; √12, 4, 2√3; √18, 9, 3√2; √50, 25, 5√2; √75, 25, 5√3.SurdFactorSimplified√842√2√1242√3√1893√2√50255√2√75255√3
Each surd with the largest square factor of its number, and its simplest form.

Now add. Surds with the same number under the root are like surds. Add them as you add like things: 2 apples and 3 apples make 5 apples. Simplify first, to see whether the surds are alike.

Worked example

Simplify √8 + 3√2.

  1. Simplify √8 first: √8 = √4 × √2 = 2√2.
  2. Now both terms have √2: 2√2 + 3√2.
  3. Add the numbers in front, as with apples: 2 + 3 = 5.
  4. So √8 + 3√2 = 5√2.

Answer: √8 + 3√2 = 5√2

To multiply surds, use √a × √b = √(a × b), then simplify the result.

Worked example

Simplify √6 × √3.

  1. Use √a × √b = √(a × b). So √6 × √3 = √(6 × 3).
  2. 6 × 3 = 18, so the product is √18.
  3. Find a square factor of 18: 18 = 9 × 2.
  4. √18 = √9 × √2, and √9 = 3.
  5. So √6 × √3 = 3√2.

Answer: √6 × √3 = 3√2

A surd on the bottom of a fraction is awkward. To rationalise the denominator means to change the fraction so the bottom is a whole number. Multiply the top and the bottom by the same surd. That is multiplying by 1, so the value does not change.

Worked example

Rationalise the denominator of 6/√3.

  1. The bottom is √3. Multiply the top and the bottom by √3. This is the same as multiplying by 1.
  2. Bottom: √3 × √3 = √9 = 3.
  3. Top: 6 × √3 = 6√3. So the fraction is 6√3 ÷ 3.
  4. Divide the top number by the bottom: 6 ÷ 3 = 2.
  5. So 6/√3 = 2√3.
  6. Check: √3 is about 1.732. Then 6 ÷ 1.732 is about 3.464, and 2 × 1.732 = 3.464.

Answer: 6/√3 = 2√3

See it three ways

With things

Think of √2 as one stick. Lay out 2 sticks and then 3 more sticks. Count the sticks.

As a picture

Two groups of tiles, each tile standing for one √2. The first group has 2 dotted tiles, with the label 2√2 under it. The second group has 3 plain tiles, with the label 3√2 under it.2√23√2

In symbols

2√2 + 3√2 = 5√2, the way 2 + 3 = 5.

Try it

Check with a calculator

You need: a calculator, or paper and a pencil.

  1. Use √3 = 1.732, which is about right.
  2. Work out 6 ÷ 1.732.
  3. Work out 2 × 1.732 and compare.

What you will see: 6 ÷ 1.732 is about 3.464, and 2 × 1.732 = 3.464. They agree, so 6/√3 and 2√3 are the same number.

If you cannot try it, this is what happens.
A table with the head Sum and About. Rows: 6 ÷ √3, 3.464; 2√3, 3.464.SumAbout6 ÷ √33.4642√33.464

A wrong idea many people have

√9 + √16 = √25, because you can add the numbers under the roots.

Why it feels right: The roots look like numbers that can be added inside, the way brackets work.

The test that breaks it: Work both sides out. √9 + √16 = 3 + 4 = 7, but √25 = 5. They are not equal.

What is true: You can split a product under a root, √(a × b) = √a × √b. You cannot split a sum: √(a + b) is not √a + √b.

A wrong idea many people have

√50 = 25√2, because 25 is the square factor.

Why it feels right: The 25 is the number that divides 50, so it seems to come out in front.

The test that breaks it: Check with decimals. √50 is about 7.07. But 25√2 is about 35.4.

What is true: Only the square root of the square factor comes out. √25 = 5, so √50 = 5√2, which is about 7.07.

Key words

surd
an exact root that is not a whole number or fraction, such as √2.
square factor
a factor of a number that is itself a square number, such as 25 for 50.
rationalise
change a fraction so that its bottom is a whole number, not a surd.

Back to your guess

The answer is C: √50 = 5√2, because 50 = 25 × 2 and √25 = 5.

50 = 25 × 2, and 25 is a square number with √25 = 5. The 5 comes out of the root sign and the 2 stays under it. So √50 = 5√2. It is not 25√2, because only the square root of 25 comes out.

Check yourself

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

  1. Simplify √72.

    Tap your answer.

    1. Not this one. You took 36 out in front. Only the square root of the square factor comes out: √36 = 6, not 36. Go to M4.1.7. Try another.

    2. Yes, that is right. 72 = 36 × 2, so √72 = √36 × √2 = 6√2.

    3. Not this one. 72 has a square factor, 36, because 72 = 36 × 2. So it can be simplified. Go to M4.1.7. Try another.

    See the answer

    6√2

    72 = 36 × 2, so √72 = √36 × √2 = 6√2.

    • If you chose A: You took 36 out in front. Only the square root of the square factor comes out: √36 = 6, not 36. Go to M4.1.7.
    • If you chose C: 72 has a square factor, 36, because 72 = 36 × 2. So it can be simplified. Go to M4.1.7.

  2. Simplify √12 + √27.

    Tap your answer.

    1. Not this one. You added the numbers under the roots. Check: √12 + √27 is about 3.46 + 5.20 = 8.66, but √39 is only about 6.24. Go to M4.1.7. Try another.

    2. Not this one. The number under the root stays the same when you add like surds. Only the numbers in front are added. Go to M4.1.7. Try another.

    3. Yes, that is right. √12 = 2√3 and √27 = 3√3. Both are like surds, so 2√3 + 3√3 = 5√3.

    See the answer

    5√3

    √12 = 2√3 and √27 = 3√3. Both are like surds, so 2√3 + 3√3 = 5√3.

    • If you chose A: You added the numbers under the roots. Check: √12 + √27 is about 3.46 + 5.20 = 8.66, but √39 is only about 6.24. Go to M4.1.7.
    • If you chose B: The number under the root stays the same when you add like surds. Only the numbers in front are added. Go to M4.1.7.

  3. Work out √2 × √8.

    Tap your answer.

    1. Yes, that is right. √2 × √8 = √(2 × 8) = √16 = 4.

    2. Not this one. You added 2 and 8. For a product, multiply the numbers under the root: 2 × 8 = 16. Go to M4.1.7. Try another.

    3. Not this one. You multiplied 2 and 8 but did not take the root. √16 = 4. Go to M4.1.7. Try another.

    See the answer

    4

    √2 × √8 = √(2 × 8) = √16 = 4.

    • If you chose B: You added 2 and 8. For a product, multiply the numbers under the root: 2 × 8 = 16. Go to M4.1.7.
    • If you chose C: You multiplied 2 and 8 but did not take the root. √16 = 4. Go to M4.1.7.

  4. Rationalise the denominator of 10/√5.

    Tap your answer.

    1. Not this one. You multiplied the top by √5 but forgot that the bottom becomes 5. Divide the top number by 5 as well: 10 ÷ 5 = 2. Go to M4.1.7. Try another.

    2. Yes, that is right. Multiply the top and bottom by √5. The bottom becomes √5 × √5 = 5, and the top becomes 10√5. Then 10√5 ÷ 5 = 2√5.

    3. Not this one. You cancelled 10 with 5 as if the bottom were 5. The bottom was √5. The root stays in the answer. Go to M4.1.7. Try another.

    See the answer

    2√5

    Multiply the top and bottom by √5. The bottom becomes √5 × √5 = 5, and the top becomes 10√5. Then 10√5 ÷ 5 = 2√5.

    • If you chose A: You multiplied the top by √5 but forgot that the bottom becomes 5. Divide the top number by 5 as well: 10 ÷ 5 = 2. Go to M4.1.7.
    • If you chose C: You cancelled 10 with 5 as if the bottom were 5. The bottom was √5. The root stays in the answer. Go to M4.1.7.

  5. Tariq says √36 + √64 = √100. Is he right?

    Tap your answer.

    1. Yes, that is right. A root of a sum is not the sum of the roots. Only a root of a product splits.

    2. Not this one. Try it with numbers: √36 + √64 = 6 + 8 = 14, but √100 = 10. Adding under the root is not allowed. Go to M4.1.7. Try another.

    3. Not this one. 36 and 64 are square numbers, and it still fails: 14 is not 10. Go to M4.1.7. Try another.

    See the answer

    No. √36 + √64 = 6 + 8 = 14, but √100 = 10.

    A root of a sum is not the sum of the roots. Only a root of a product splits.

    • If you chose B: Try it with numbers: √36 + √64 = 6 + 8 = 14, but √100 = 10. Adding under the root is not allowed. Go to M4.1.7.
    • If you chose C: 36 and 64 are square numbers, and it still fails: 14 is not 10. Go to M4.1.7.

  6. From an earlier chapter

    Write 0.131313... as a fraction in its simplest form.

    See the answer

    13/99

    The repeat 13 has 2 digits, so 99x = 13 and x = 13/99. It cannot be simplified.

  7. Work out √7 × √7.

    See the answer

    7

    √7 × √7 = √(7 × 7) = √49 = 7.

  8. Simplify √18 + √8. What is the number in front of the root?

    See the answer

    5

    √18 = 3√2 and √8 = 2√2. Like surds add: 3 + 2 = 5, so the answer is 5√2.

  9. Rationalise the denominator of 12/√6. What is the number in front of the root?

    See the answer

    2

    Multiply the top and bottom by √6. The bottom becomes 6 and the top becomes 12√6. Then 12 ÷ 6 = 2, so the answer is 2√6.

Keep it

Copy this into your exercise book.

Find the biggest square factor: √50 = √25 × √2 = 5√2. Like surds add: 2√2 + 3√2 = 5√2. For a fraction, multiply top and bottom by the surd: 6/√3 = 2√3.

Draw: A factor tree for 50 with a ringed pair of 5s, and √50 = 5√2 written beside it.

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

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