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Welcome to our Math lesson on Multiplying Brackets containing Surds, this is the first lesson of our suite of math lessons covering the topic of Rationalising the Denominator, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Multiplying Brackets containing Surds
In tutorials 6.3 and 6.4 we discussed the eight special algebraic identities that help us factorise or simplify various expressions. All those identities are also useful when dealing with surds, especially the first three of them, i.e.
(a + b)2 = a2 + 2ab + b2
(a-b)2 = a2 -2ab + b2
and
(a-b)(a + b) = a2 -b2
The third special identity is particularly useful as it allows us getting rid of surds immediately, without the need for further operations. For example, in the expression
(5-√2)(5 + √2)
we can get rid of roots quickly, i.e.
(5 - √2)(5 + √2)
= 52 - (√2)2
= 25 - 2
= 23
However, the first two special algebraic identities are also useful as if we have let's say the sum or difference of two surds raised in the second power, the result will contain a single surd. For example,
(√3 + √2)2
= (√3)2 + 2 ∙ √3 ∙ √2 + (√2)2
= 3 + 2√(3 ∙ 2) + 2
= 5 + 2√6
The rest of algebraic identities are also useful in some situations. For example, the expression
11√2 + 9√3
is the simplified form of
(√2 + √3)3
because given the fourth algebraic identity (a + b)3 = a3 + 3a2b + 3ab2 + b3, we can write
(√2 + √3)3
= (√2)3 + 3 ∙ (√2)2 ∙ √3 + 3 ∙ √2 ∙ (√3)2 + (√3)3
= (√2)2 ∙ √2 + 3 ∙ (√2)2 ∙ √3 + 3 ∙ √2 ∙ (√3)2 + (√3)2 ∙ √3
= 2√2 + 3 ∙ 2 ∙ √3 + 3 ∙ √2 ∙ 3 + 3√3
= 2√2 + 6√3 + 9√2 + 3√3
= 11√2 + 9√3
Example 1
Write the following expressions in the simplest form.
- (3 - √7)(3 + √7)
- (√6 - 1)2 + (√2 + √3)2
Solution 1
- Applying the formula of the third special algebraic identity, yields:
(3 - √7)(3 + √7)
= 32 - (√7)2
= 9 - 7
= 2
- Combining the formulae of the first two special algebraic identities yields
(√6 - 1)2 + (√2 + √3)2
= (√6)2 - 2 ∙ √6 ∙ 1 + 12 + (√2)2 + 2 ∙ √2 ∙ √3 + (√3)2
= 6 - 2√6 + 1 + 2 + 2√6 + 3
= 12
More Rationalising the Denominator Lessons and Learning Resources
Powers and Roots Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 7.5 | Rationalising the Denominator | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 7.5.1 | Multiplying Brackets containing Surds | | |
| 7.5.2 | Rationalising the Denominator | | |
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