This worksheet guides students through simplifying radicals, expanding brackets, and squaring surd expressions.

Let’s solve each problem step by step. We’ll go section by section, simplifying radicals and expanding expressions carefully.

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Section A: Simplify the following radicals

We combine like terms — that means only add or subtract radicals with the same number under the root (√).

1) 5√2 + 7√2 = (5+7)√2 = 12√2

2) 7√2 – 4√2 + 9√3 – 2√3
= (7–4)√2 + (9–2)√3 = 3√2 + 7√3

3) 11√5 + 3√7 + 4√5 + 6√7
= (11+4)√5 + (3+6)√7 = 15√5 + 9√7

4) 16√5 + 9√7 – 7√5 – 4√7
= (16–7)√5 + (9–4)√7 = 9√5 + 5√7

5) 3√2 + 7√3 – 7√3 – 8√2
= (3–8)√2 + (7–7)√3 = –5√2 + 0 = –5√2

Now for problems 6–10, we need to simplify the radicals first before combining.

6) √18 + √50
√18 = √(9×2) = 3√2
√50 = √(25×2) = 5√2
So: 3√2 + 5√2 = 8√2

7) √72 – √8
√72 = √(36×2) = 6√2
√8 = √(4×2) = 2√2
So: 6√2 – 2√2 = 4√2

8) 4√12 + 2√27
√12 = √(4×3) = 2√3 → 4×2√3 = 8√3
√27 = √(9×3) = 3√3 → 2×3√3 = 6√3
So: 8√3 + 6√3 = 14√3

9) 5√28 – 3√63
√28 = √(4×7) = 2√7 → 5×2√7 = 10√7
√63 = √(9×7) = 3√7 → 3×3√7 = 9√7
So: 10√7 – 9√7 = √7

10) 2√45 + 5√20 – 4√98
√45 = √(9×5) = 3√5 → 2×3√5 = 6√5
√20 = √(4×5) = 2√5 → 5×2√5 = 10√5
√98 = √(49×2) = 7√2 → 4×7√2 = 28√2
So: 6√5 + 10√5 – 28√2 = 16√5 – 28√2

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Section B: Expand the parentheses and simplify the radicals

Use distributive property (multiply outside term by each inside term), then simplify.

1) √3(2 – √3) = 2√3 – (√3 × √3) = 2√3 – 3 → –3 + 2√3

2) 2√7(7 – √7) = 2√7×7 – 2√7×√7 = 14√7 – 2×7 = 14√7 – 14 → –14 + 14√7

3) √5(5√5 – √3) = √5×5√5 – √5×√3 = 5×5 – √15 = 25 – √15 → 25 – √15

4) 2√11(4 – √11) = 2√11×4 – 2√11×√11 = 8√11 – 2×11 = 8√11 – 22 → –22 + 8√11

5) (√7 + 3)(√7 – 3) → This is difference of squares: (a+b)(a-b) = a² – b²
= (√7)² – (3)² = 7 – 9 = –2

6) (8 + √5)(8 – √5) → Again, difference of squares
= 8² – (√5)² = 64 – 5 = 59

7) (5 + √3)(4 + 2√3)
Multiply using FOIL:
First: 5×4 = 20
Outer: 5×2√3 = 10√3
Inner: √3×4 = 4√3
Last: √3×2√3 = 2×3 = 6
Add: 20 + 6 + 10√3 + 4√3 = 26 + 14√3 → 26 + 14√3

8) (7√5 + 2)(√5 + 4)
FOIL:
First: 7√5×√5 = 7×5 = 35
Outer: 7√5×4 = 28√5
Inner: 2×√5 = 2√5
Last: 2×4 = 8
Add: 35 + 8 + 28√5 + 2√5 = 43 + 30√5 → 43 + 30√5

9) (1 – 3√7)(6 – 3√7)
FOIL:
First: 1×6 = 6
Outer: 1×(-3√7) = -3√7
Inner: -3√7×6 = -18√7
Last: (-3√7)×(-3√7) = 9×7 = 63
Add: 6 + 63 + (-3√7 -18√7) = 69 – 21√7 → 69 – 21√7

10) (√3 – 1)(9√3 + 5)
FOIL:
First: √3×9√3 = 9×3 = 27
Outer: √3×5 = 5√3
Inner: -1×9√3 = -9√3
Last: -1×5 = -5
Add: 27 – 5 + 5√3 – 9√3 = 22 – 4√3 → 22 – 4√3

11) (4√7 + 3)(3√7 – 1)
FOIL:
First: 4√7×3√7 = 12×7 = 84
Outer: 4√7×(-1) = -4√7
Inner: 3×3√7 = 9√7
Last: 3×(-1) = -3
Add: 84 – 3 + (-4√7 + 9√7) = 81 + 5√7 → 81 + 5√7

12) (4√3 – √2)(3√3 – 4√2)
FOIL:
First: 4√3×3√3 = 12×3 = 36
Outer: 4√3×(-4√2) = -16√6
Inner: -√2×3√3 = -3√6
Last: -√2×(-4√2) = 4×2 = 8
Add: 36 + 8 + (-16√6 – 3√6) = 44 – 19√6 → 44 – 19√6

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Section C: Expand the parentheses and simplify the radicals

These involve squaring binomials or more complex expansions.

Recall: (a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²

1) (√2 + 1)² = (√2)² + 2×√2×1 + 1² = 2 + 2√2 + 1 = 3 + 2√2

2) (√11 – 2)² = (√11)² – 2×√11×2 + 2² = 11 – 4√11 + 4 = 15 – 4√11

3) (3 – √5)² = 3² – 2×3×√5 + (√5)² = 9 – 6√5 + 5 = 14 – 6√5

4) √22(√3 + √7)²
First expand (√3 + √7)² = (√3)² + 2√3√7 + (√7)² = 3 + 2√21 + 7 = 10 + 2√21
Now multiply by √22: √22×(10 + 2√21) = 10√22 + 2√(22×21) = 10√22 + 2√462
Can’t simplify √462 further (462 = 2×3×7×11, no square factors) → 10√22 + 2√462

5) √11(√11 – √6)²
First expand (√11 – √6)² = (√11)² – 2√11√6 + (√6)² = 11 – 2√66 + 6 = 17 – 2√66
Now multiply by √11: √11×(17 – 2√66) = 17√11 – 2√(11×66)
But 11×66 = 726 → √726 = √(121×6) = 11√6
So: 17√11 – 2×11√6 = 17√11 – 22√6 → 17√11 – 22√6

6) ((√2)³ + √5)²
First, (√2)³ = (√2)² × √2 = 2√2
So expression becomes: (2√2 + √5)²
Now expand: (2√2)² + 2×2√2×√5 + (√5)² = 4×2 + 4√10 + 5 = 8 + 5 + 4√10 = 13 + 4√10

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Final Answer:

Section A
1) 12√2
2) 3√2 + 7√3
3) 15√5 + 9√7
4) 9√5 + 5√7
5) –5√2
6) 8√2
7) 4√2
8) 14√3
9) √7
10) 16√5 – 28√2

Section B
1) –3 + 2√3
2) –14 + 14√7
3) 25 – √15
4) –22 + 8√11
5) –2
6) 59
7) 26 + 14√3
8) 43 + 30√5
9) 69 – 21√7
10) 22 – 4√3
11) 81 + 5√7
12) 44 – 19√6

Section C
1) 3 + 2√2
2) 15 – 4√11
3) 14 – 6√5
4) 10√22 + 2√462
5) 17√11 – 22√6
6) 13 + 4√10