Section A
Worked Example
𝟒𝟒 𝟐𝟐
Calculate +
𝟕𝟕 𝟑𝟑
Step 1: Manipulate each fraction so that they have the same denominator (bottom number).
To do this, we need to find the lowest common multiple of both denominators. Here, it is 21.
To change a fraction, we need to multiply the numerator and denominator by the same
number.
4 3 12
× =
7 3 21
2 7 14
× =
3 7 21
Step 2: Perform the operation on the fractions that have the same denominator.
12 14 26
+ =
21 21 21
Step 3: Simplify the fraction if possible and leave in its exact form.
This fraction cannot be simplified, so we leave it as
𝟐𝟐𝟐𝟐
𝟐𝟐𝟐𝟐
Guided Example
𝟏𝟏𝟏𝟏 𝟑𝟑
Calculate −
𝟏𝟏𝟏𝟏 𝟓𝟓
Step 1: Manipulate each fraction so that they have the same denominator (bottom number).
Step 2: Perform the operation on the fractions that have the same denominator.
Step 3: Simplify the fraction if possible and leave in its exact form.
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1. Calculate the following, leaving your answer in exact form.
5 1
a) +
9 3
1 4
b) 1 −
2 7
2 3
c) 3 + 2
3 4
3 7
d) −
8 10
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Section B
Worked Example
𝟑𝟑 𝟏𝟏
Calculate ×
𝟕𝟕 𝟐𝟐
Step 1: To multiply exact fractions, multiply the numerators of each, and multiply the denominators
of each.
3 1 3×1 3
× = =
7 2 7 × 2 14
Step 2: Write the final fraction in exact form and check if it can be simplified.
The final answer cannot be simplified, so we leave the answer as
𝟑𝟑
𝟏𝟏𝟏𝟏
Guided Example
𝟑𝟑 𝟔𝟔
Calculate ×
𝟒𝟒 𝟕𝟕
Step 1: To multiply exact fractions, multiply the numerators of each, and multiply the denominators
of each.
Step 2: Write the final fraction in exact form and check if it can be simplified.
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2. Calculate the following, leaving your answer in exact form.
5 2
a) ×
6 3
1 7
b) 2 ×
4 9
−4 2
c) ×1
5 7
2 2
d) 4 ×
3 3
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Section C
Worked Example
𝟑𝟑 𝟏𝟏
Calculate ÷
𝟕𝟕 𝟔𝟔
Step 1: To divide exact fractions, we need to flip the numerator and denominator of the second
fraction.
6
Flipping the second fraction gives us .
1
Step 2: We then change the sign from division to multiplication and multiply the fractions (multiply
the numerators and denominators).
3 6 18
× =
7 1 7
Step 3: Simplify the fraction if possible and leave in its exact form.
The final answer cannot be simplified, so we leave it as:
𝟏𝟏𝟏𝟏
𝟕𝟕
Guided Example
𝟔𝟔 𝟓𝟓
Calculate ÷
𝟕𝟕 𝟑𝟑
Step 1: To divide exact fractions, we need to flip the numerator and denominator of the second
fraction.
Step 2: We then change the sign from division to multiplication and multiply the fractions (multiply
the numerators and denominators).
Step 3: Simplify the fraction if possible and leave in its exact form.
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3. Calculate the following, leaving your answer in exact form.
2 1
a) ÷
3 7
5 4
b) ÷
6 3
9 5
c) ÷
11 3
15 3
d) ÷
4 9
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Section D – Higher Only
Worked Example
Calculate 𝟐𝟐√𝟓𝟓 + 𝟐𝟐√𝟓𝟓
Step 1: We can only add together surds if they have the same number under the square root.
Identify which surds are the same.
Both terms here have 5 under the square root, so they can be added.
Step 2: Look at the number outside the surd. This tells us the multiple of that surd. Use these
numbers to add together surds with the same number under the square root.
Both surds have a 2 outside the square root, meaning 2 × √5. If we were to write out this
calculation in full, it would be √5 + √5 + √5 + √5. Collect the terms that are the same.
This gives us 4√5.
Guided Example
Calculate 𝟓𝟓√𝟕𝟕 − 𝟐𝟐√𝟕𝟕
Step 1: We can only add together surds if they have the same number under the square root.
Identify which surds are the same.
Step 2: Look at the number outside the surd. This tells us the multiple of that surd. Use these
numbers to subtract the surds with the same number under the square root.
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4. Calculate the following, leaving your answer as a surd.
a) 10√6 − √6
b) 2√3 + √5 + 3√3 + 4√5
c) 3√6 − 5√6
d) 6√8 + 3√3 − 2√8 + 4√3
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Section E – Higher Only
Worked Example
Simplify the surd √𝟏𝟏𝟏𝟏𝟏𝟏
Step 1: Find the largest square number that is a factor of the number under the square root.
125 = 𝟐𝟐𝟐𝟐 × 5
25 is a square number factor (25 = 52 )
Step 2: Write the factors under the square root, then split into two surds.
√125 = √25 × 5 = √25 × √5
Step 3: Simplify the surd that is a square number and write the final surd.
√25 = 5
So,
√125 = √25 × 5 = √25 × √5 = 5 × √5 = 𝟓𝟓√𝟓𝟓
Guided Example
Simplify the surd 𝟐𝟐√𝟏𝟏𝟏𝟏
Step 1: Find the largest square number that is a factor of the number under the square root.
Step 2: Write the factors under the square root, then split into two surds.
Step 3: Simplify the surd that is a square number and write the final surd.
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5. Simplify the following surds:
a) √72
b) 6√12
c) 5√8 + 6√28
d) 4√4 + 6√16
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Section F – Higher Only
Worked Example
Calculate 𝟒𝟒√𝟕𝟕 × √𝟖𝟖
Step 1: Check if the surds can be simplified.
√8 = √4 × 2 = √4 × √2 = 2√2
So, the calculation simplifies to: 4√7 × 2√2
Step 2: When multiplying or dividing surds, perform the operation on the numbers under the square
root and the numbers outside separately.
We are now calculating: 4√7 × 2√2
4√7 × 2√2 = (4 × 2)√7 × 2 = 8√14
Step 3: Write the final surd, simplifying again if possible.
The final answer is 𝟖𝟖√𝟏𝟏𝟏𝟏, which cannot be simplified further.
Guided Example
𝟒𝟒√𝟔𝟔
Calculate
𝟐𝟐√𝟏𝟏𝟏𝟏
Step 1: First, check if the surds can be simplified.
Step 2: When multiplying or dividing surds, perform the operation on the numbers under the square
root and the numbers outside separately.
Step 3: Write the final surd, simplifying again if possible.
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6. Calculate the following:
a) √105 ÷ √15
b) 10√3 × 2√27
15√10
c)
3√2
d) −2√12 × 4√12
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Section G – Higher Only
Worked Example
𝟓𝟓
Rationalise the denominator of the fraction
√𝟓𝟓
Step 1: Identify the surd in the denominator.
Looking at the bottom of the fraction, we see the surd present is √5.
Step 2: Multiply the numerator and denominator by this surd.
When we multiply the denominator by √5, we are squaring a surd, which removes the
square root and makes it an integer.
We have to multiply the numerator and denominator by the same number, because this is
the same as multiplying it by 1.
5 √5 5√5
× =
√5 √5 5
Step 3: Write the final fraction, simplifying if possible.
5√5
= √𝟓𝟓
5
Guided Example
𝟏𝟏𝟏𝟏
Rationalise the denominator of the fraction
𝟐𝟐√𝟔𝟔
Step 1: Identify the surd in the denominator.
Step 2: Multiply the numerator and denominator by this surd.
Step 3: Write the final fraction, simplifying if possible.
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7. Rationalise the denominators of the following fractions:
3
a)
4√7
10√12
b)
3√6
4√10
c) −
4√16
9√2
d)
18√3
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