AQA · Mathematics · 8300
GCSE Maths: Surds: expanding brackets
Algebra · AQA GCSE Maths (8300). Cards are generated from the Corbettmaths textbook exercise and practice questions for Surds: expanding brackets (Corbettmaths Video 308), and every card links back to the exact page it came from. Practice tests are drawn from this deck’s own cards. Source material © Corbettmaths — used with attribution.
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What is √a × √b equal to?
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√(a×b). For example, √3 × √5 = √15.
When you multiply a surd by itself, (√a)² = ?
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a
What does √a ÷ √b simplify to?
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√(a÷b). For example, √10 ÷ √5 = √2.
How do you simplify √50?
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Find the largest square factor: √50 = √(25×2) = 5√2.
When can you add or subtract surds directly?
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Only when they have the same surd part. For example, 2√3 + 5√3 = 7√3, but √2 + √3 cannot be simplified.
How do you expand √2(√3 + 5)?
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Distribute: √2 × √3 + √2 × 5 = √6 + 5√2.
What is the result of expanding (a + √b)(a − √b)?
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a² − b (difference of two squares). For example, (4 + √3)(4 − √3) = 16 − 3 = 13.
What does it mean to rationalise a denominator?
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To remove any surds from the denominator by multiplying both numerator and denominator by an appropriate surd.
To expand (a + √b)², use the pattern a² + 2a√b + ?
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b
What is (√3)³ equal to?
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(√3)³ = √3 × √3 × √3 = 3√3.
Is √2 + √8 equal to √10?
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No. √2 + √8 = √2 + 2√2 = 3√2, not √10. You cannot add square roots by adding the numbers inside.
Rationalise the denominator of 15/√5.
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3√5. Multiply top and bottom by √5: (15√5)/(√5 × √5) = 15√5/5 = 3√5.
Expand and simplify (3 + √5)².
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14 + 6√5. Using (a+b)² = a² + 2ab + b²: 9 + 2(3)(√5) + 5 = 9 + 6√5 + 5 = 14 + 6√5.
Evaluate (5 + √2)(5 − √2).
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23. Using the difference of squares (a+b)(a−b) = a² − b²: 25 − 2 = 23.
Simplify √3(√27 − √3).
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6. √3 × √27 = √81 = 9, and √3 × √3 = 3, so 9 − 3 = 6.
Expand (8 − √3)² and give your answer in the form a + b√3.
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67 − 16√3. Using (a−b)² = a² − 2ab + b²: 64 − 16√3 + 3 = 67 − 16√3.
Simplify √80 − 10/√5.
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2√5. √80 = 4√5, and 10/√5 = 10√5/5 = 2√5, so 4√5 − 2√5 = 2√5.
Rationalise the denominator of 6/(2 − √3).
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12 + 6√3. Multiply top and bottom by the conjugate (2 + √3): 6(2+√3)/[(2−√3)(2+√3)] = (12+6√3)/(4−3) = 12 + 6√3.
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