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10&10a

2.04 Binomial expansions with surds

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Binomial expansions with surds

We know how to  multiply surds  together using the rule \sqrt{a} \times \sqrt{b} =\sqrt{ab}\,, and we have also looked at how to  expand binomial products  using the rule (A+B)(C+D)=AC+AD+BC+BD.

Now, there is nothing special about A,\,B,\,C, or D. They can be any terms, and needn't be a variable or an integer. This means we can combine the two concepts together allowing us to expand binomial expressions involving surds.

Examples

Example 1

Expand and simplify: \left(5-\sqrt{7}\right)\left(8-\sqrt{3}\right)

Worked Solution
Create a strategy

Expand using the distributive law: (A-B)(C-D)=A(C-D)-B(C-D)

Apply the idea
\displaystyle \left(5-\sqrt{7}\right)\left(8-\sqrt{3}\right)\displaystyle =\displaystyle 5\left(8-\sqrt{3}\right)-\sqrt{7}\left(8-\sqrt{3}\right)Apply the distributive law
\displaystyle =\displaystyle 40-5\sqrt{3}-8\sqrt{7}+\sqrt{21}Expand both brackets

Example 2

Expand and simplify: \left(10\sqrt{2}-10\right)\left(10\sqrt{2}+10\right)

Worked Solution
Create a strategy

Expand using the rule for the difference of two squares: \left(A+B\right)\left(A-B\right)=A^{2}-B^{2}

Apply the idea
\displaystyle \left(10\sqrt{2}-10\right)\left(10\sqrt{2}+10\right)\displaystyle =\displaystyle \left(10\sqrt{2}\right)^{2}-10^{2}Apply the rule
\displaystyle =\displaystyle 100 \times 2-100Evaluate the exponents
\displaystyle =\displaystyle 100Evaluate

Example 3

Expand and simplify: \left(3\sqrt{11}-\sqrt{5}\right)^{2}

Worked Solution
Create a strategy

Use the binomial expansion \left(A-B \right)^{2}=A^{2}-2AB+B^{2} to expand the brackets.

Apply the idea
\displaystyle \left(3\sqrt{11}-\sqrt{5}\right)^{2}\displaystyle =\displaystyle \left(3\sqrt{11}\right)^{2}-2\times 3\sqrt{11} \times \sqrt{5}+\left(\sqrt{5}\right)^{2}Apply the rule
\displaystyle =\displaystyle 9 \times 11-6\sqrt{55} +5Evaluate the exponents and product
\displaystyle =\displaystyle 99-6\sqrt{55}+5Perform the multiplication
\displaystyle =\displaystyle 104-6\sqrt{55}Add the integer terms
Idea summary

We can expand the product of two binomial expressions using the rule: (A+B)(C+D)=AC+AD+BC+BDThere are two special cases of expanding binomials:

  • \left(A+B \right)^{2}=A^{2}+2AB+B^{2} (called a perfect square)
  • \left(A+B\right)\left(A-B\right)=A^{2}-B^{2} (called a difference of two squares)

Outcomes

ACMNA264 (10a)

Define rational and irrational numbers and perform operations with surds and fractional indices

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