29 lines
2.2 KiB
Markdown
29 lines
2.2 KiB
Markdown
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---
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title: Simplifying Square Roots
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---
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### Example: Simplify <img src="https://latex.codecogs.com/gif.latex?\sqrt{180}" title="\sqrt{180}" />
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Solution steps:
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* Break 180 into product of prime number factors: <img src="https://latex.codecogs.com/gif.latex?180&space;=&space;2^2&space;*&space;3^2&space;*&space;5" title="180 = 2^2 * 3^2 * 5" />
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* Use product property of radicals <img src="https://latex.codecogs.com/gif.latex?\sqrt{ab}&space;=&space;\sqrt{a}\sqrt{b}&space;\Rightarrow&space;\sqrt{180}&space;=&space;\sqrt{2^2}\sqrt{3^2}\sqrt{5}=2*3\sqrt{5}=6\sqrt{5}" title="\sqrt{ab} = \sqrt{a}\sqrt{b} \Rightarrow \sqrt{180} = \sqrt{2^2}\sqrt{3^2}\sqrt{5}=2*3\sqrt{5}=6\sqrt{5}" />
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We cannot simplify square root of a prime number thus final answer: <img src="https://latex.codecogs.com/gif.latex?\sqrt{180}&space;=&space;6\sqrt{5}" title="\sqrt{180} = 6\sqrt{5}" />
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### Example: Simplify <img src="https://latex.codecogs.com/gif.latex?\sqrt{1750}" title="\sqrt{1750}" />
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Solution steps:
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* Break 1750 into product of prime number factors: <img src="https://latex.codecogs.com/gif.latex?1750&space;=&space;2*7*5^3" title="1750 = 2*7*5^3" />
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* Use product property of radicals and keep breaking down the results: <img src="https://latex.codecogs.com/gif.latex?\sqrt{ab}&space;=&space;\sqrt{a}\sqrt{b}&space;\Rightarrow&space;\sqrt{1750}&space;=&space;\sqrt{2}\sqrt{7}\sqrt{5^3}&space;=&space;\sqrt{2}\sqrt{7}\sqrt{5}\sqrt{5^2}&space;=&space;5\sqrt{2}\sqrt{7}\sqrt{5}" title="\sqrt{ab} = \sqrt{a}\sqrt{b} \Rightarrow \sqrt{1750} = \sqrt{2}\sqrt{7}\sqrt{5^3} = \sqrt{2}\sqrt{7}\sqrt{5}\sqrt{5^2} = 5\sqrt{2}\sqrt{7}\sqrt{5}" />
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Now use product property again to combine radicals which cannot be simplified: <img src="https://latex.codecogs.com/gif.latex?5\sqrt{2}\sqrt{7}\sqrt{5}&space;=&space;5\sqrt{70}" title="5\sqrt{2}\sqrt{7}\sqrt{5} = 5\sqrt{70}" />
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### Terminology:
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Using √49 as an example:
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* "√" is called **radical symbol**.
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* 49 is called **radicand**.
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#### More Information:
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* [Wikihow, simplifying a square root](https://www.wikihow.com/Simplify-a-Square-Root)
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* [Khan Academy](https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/alg1-simplify-square-roots/a/simplifying-square-roots-review)
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