Expanding square root brackets involves multiplying expressions containing square roots and simplifying the result. The process is similar to expanding algebraic expressions, but with the added consideration of simplifying surds (radical expressions). Here's a step-by-step guide based on the provided references:
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Simplify the surds if possible.
- Before expanding, check if any of the square roots within the brackets can be simplified. For example, √8 can be simplified to 2√2.
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Multiply each term inside the first bracket by each term inside the second bracket.
- This is the same as using the distributive property (often remembered with the acronym FOIL – First, Outer, Inner, Last – when dealing with two terms in each bracket). Each term in the first set of brackets multiplies each term in the second set of brackets.
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Collect like terms and simplify the answer.
- After multiplying, combine any like terms (terms with the same square root). Also, simplify any remaining square roots and combine any whole numbers.
Example: Expanding (√2 + 1)(√2 - 1)
Let's illustrate this with an example: Expand (√2 + 1)(√2 - 1).
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Step 1: Simplify Surds. In this case, √2 is already in its simplest form.
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Step 2: Multiply each term.
(√2 + 1)(√2 - 1) = (√2 √2) + (√2 -1) + (1 √2) + (1 -1)
= 2 - √2 + √2 - 1
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Step 3: Collect like terms and simplify.
= 2 - 1 - √2 + √2
= 1 + 0
= 1
Another Example: Expanding (√3 + 2)(√3 + 1)
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Step 1: Simplify Surds. In this case, √3 is already in its simplest form.
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Step 2: Multiply each term.
(√3 + 2)(√3 + 1) = (√3 √3) + (√3 1) + (2 √3) + (2 1)
= 3 + √3 + 2√3 + 2
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Step 3: Collect like terms and simplify.
= 3 + 2 + √3 + 2√3
= 5 + 3√3
Tips for Expanding Square Root Brackets:
- Be careful with signs: Pay close attention to positive and negative signs when multiplying.
- Simplify early: Simplifying surds before multiplying often makes the subsequent steps easier.
- Practice: The more you practice, the more comfortable you will become with expanding and simplifying expressions containing square roots.
