Year 10 Standard · Unit 1 Number sets, surds and accuracy · Lesson 4 of 6

How are √8 and √2 related?

A surd is a root left exact, such as √2. Roots multiply, √a × √b = √(ab), so a square factor can come out of the root, and surds with the same root then add like terms.

Number and AlgebraGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: work out √144.
    Answer12, because 12² = 144.
  2. From Year 8: simplify 3x + 5x − x.
    Answer7x. Like terms are collected by adding and subtracting the numbers in front.
  3. From Year 7: list the square numbers up to 100.
    Answer1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
Words used in this lesson
surd
a square root left exact because it is irrational. For example: √2, 3√5.
square factor
a factor that is a square number. For example: 4 is a square factor of 40.
simplest form
the surd with no square factor left under the root. For example: √40 = 2√10.
like surds
surds with the same number under the root. For example: 3√2 and 5√2.
hypotenuse
the longest side of a right-angled triangle, opposite the right angle. For example: √(a + b) in the second figure.
exact
written without rounding. For example: √2, not 1.41.

A pattern of tilted tiles

A tile pattern in Isfahan uses squares drawn on a grid of dots and turned at an angle. Joining dots 1 across and 1 up gives a small tilted square with side √2. Joining dots 2 across and 2 up gives a larger one with side √8.

Predict first: is √8 twice √2, four times √2, or neither? Write down a reason, then use the first figure, set across 1 and up 1 first.

2
1
5area of the square
√5side, in simplest form

Across 2, up 1: the area is 2² + 1² = 5, so the side is √5 ≈ 2.24. 5 has no square factor except 1, so √5 cannot be simplified.

Show me the steps
  1. Across 2, up 1: the area is 2² + 1² = 5 (the box of 9 minus four triangles of 1), so the side is √5 ≈ 2.24. It cannot be simplified.
  2. Across 1, up 1: the area is 1² + 1² = 2, so the side is √2 ≈ 1.41.
  3. Across 2, up 2: the area is 8, so the side is √8 ≈ 2.83. The side passes through a grid point halfway, so it is 2 copies of the √2 side: √8 = 2√2.
  4. Across 3, up 3: the area is 18 and the side passes through 2 grid points, so it is 3 copies of √2: √18 = 3√2. Across 3, up 4 gives area 25: the side is √25 = 5, a whole number.

Roots multiply: √a × √b = √(ab), and √a ÷ √b = √(a ÷ b). So a square factor comes out of the root as its square root: √8 = √4 × √2 = 2√2. Use the largest square factor, so the number left under the root has no square factor: √128 is not finished at 2√32.

Like surds add like terms: 3√2 + 5√2 = 8√2, just as 3x + 5x = 8x. Surds with different roots, such as √2 + √3, cannot be combined, and √a + √b is never √(a + b) for positive a and b.

1
1
2√a + √b, the two legs
√2 ≈ 1.41√(a + b), the hypotenuse

Legs √1 = 1 and √1 = 1: together 2, but the hypotenuse is √2 ≈ 1.41. The legs are longer, so √1 + √1 ≠ √2.

Show me the steps
  1. a = 1, b = 1: the legs are 1 and 1, together 2. The hypotenuse is √(1 + 1) = √2 ≈ 1.41, which is shorter.
  2. a = 9, b = 16: the legs are √9 = 3 and √16 = 4, together 7. The hypotenuse is √25 = 5. So √9 + √16 is 7, not √25.
  3. a = 2, b = 8: the legs are √2 ≈ 1.41 and √8 ≈ 2.83, together about 4.24. The hypotenuse is √10 ≈ 3.16.
  4. Any a and b: the path along the two legs is longer than the straight hypotenuse, so √a + √b ≠ √(a + b) whenever a and b are positive.

Two common mistakes. Writing the square factor outside instead of its root: √40 = √4 × √10 = 2√10, not 4√10. And adding under the root: √9 + √16 = 3 + 4 = 7, but √25 = 5, so √9 + √16 is not √25.

Worked examples

1. Simplify √98.

  1. The square numbers that divide 98 are 1 and 49. The largest is 49, and 98 = 49 × 2.
  2. √98 = √49 × √2 = 7√2.

2. Simplify √6 × √15.

  1. Multiply under the root: √6 × √15 = √90.
  2. 90 = 9 × 10, so √90 = 3√10.

3. Simplify 4√3 + √75.

  1. √75 = √25 × √3 = 5√3, so both terms are now multiples of √3.
  2. 4√3 + 5√3 = 9√3.

Your go. The first step is given. Simplify √80.

  1. The largest square number that divides 80 is 16, and 80 = 16 × 5.
  2. Now take the square root of the square factor outside.
Check the answer

√80 = √16 × √5 = 4√5. Using 4 × 20 gives 2√20, which is not finished, because 20 still has the square factor 4.

With a GDC (optional)

A GDC can check a simplification. Type √98 and then 7√2: both show 9.899494937. Equal decimals to every digit shown are strong evidence that the two forms are the same number, but they are not a proof, and the exact form is what a question asks for.

Some GDCs show answers as surds; others show only a decimal. Either way, simplify by hand first, because surd questions are usually set without a calculator.

Check before you go on

1. Which of these is √12 in its simplest form?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. √45 can be written as ?√5. What number goes in front of √5?

3. Work out √3 × √12. The answer is a whole number.

Level 2 · Building

4. √72 can be written as ?√2. What number goes in front of √2?

5. √63 + 2√7 can be written as ?√7. What number goes in front of √7?

6. Work out (2√5)².

Level 3 · Stretch

7. √200 − √50 can be written as ?√2. What number goes in front of √2?

8. A rectangular garden bed in Kyoto is √12 m long and √27 m wide. What is its area, in m²? The answer is a whole number.

Something to think about

The second figure shows √a + √b ≠ √(a + b) when a and b are positive. Is it ever true? Try squaring both sides.

One way to think about it

Squaring the left side gives a + 2√(ab) + b, and the right side gives a + b. They are equal only when 2√(ab) = 0, so only when a = 0 or b = 0. For example √0 + √9 = √9. In the figure that is a triangle with one leg of length 0, which is just a line.

Exit ticket

9. Which of these is equal to √6 + √24?