Expressing square roots in their simplest radical form is necessary for several reasons:
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Clarity: Simplifying square roots makes mathematical expressions clearer and easier to understand. It allows for quicker recognition of relationships between numbers and makes calculations more straightforward.
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Standardization: In mathematics, presenting answers in a standard format facilitates communication and understanding. Simplified forms are more recognizable and universally understood.
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Practicality: In real-world applications, having numbers in their simplest form can make calculations easier, whether in further calculations, comparisons, or estimations.
Process of Rewriting a Square Root in Simplest Radical Form
To rewrite a square root in its simplest radical form, follow these steps:
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Factor the Number: Break down the number under the square root into its prime factors.
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Identify Perfect Squares: Look for any pairs of the same factor. Each pair corresponds to a perfect square that can be taken out of the square root.
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Simplify: Pull the perfect squares out of the square root and leave any leftover factors inside.
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Write the Final Expression: Combine the factors you took out of the square root with the remaining factors inside the square root.
Example
Let's simplify \(\sqrt{72}\):
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Factor the Number:
- \(72 = 36 \times 2\)
- \(36 = 6 \times 6 = 6^2\)
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Identify Perfect Squares:
- The perfect square here is \(36\).
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Simplify:
- \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\)
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Write the Final Expression:
- The simplest radical form of \(\sqrt{72}\) is \(6\sqrt{2}\).
This process can be applied consistently to simplify any square root.