Understanding Surds

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What is a surd?
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A surd is an irrational number that can’t be expressed as a simple fraction, usually involving roots such as square roots or cube roots.
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Give an example of a surd.
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The square root of 2 (\(\sqrt{2}\)) is a surd because it cannot be simplified to a rational number.
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Is \(\sqrt{4}\) a surd? Why or why not?
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No, \(\sqrt{4}\) is not a surd because it simplifies to 2, a rational number.
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Simplify \(\sqrt{50}\) into its simplest surd form.
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\(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\)
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How do you multiply surds?
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To multiply surds, multiply the numbers under the roots and then simplify if possible. For example, \(\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6\).
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Can surds be added directly like integers? Why?
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No, surds can only be added or subtracted if they have the same radicand, similar to adding like terms in algebra.
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What is the product of \(\sqrt{6} \times \sqrt{6}\)?
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The product is 6, because \(\sqrt{6} \times \sqrt{6} = (\sqrt{6})^2 = 6\).
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What is the simplest form of \(\sqrt{8} + 2\sqrt{2}\)?
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\(\sqrt{8} = 2\sqrt{2}\), so \(\sqrt{8} + 2\sqrt{2} = 4\sqrt{2}\).
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Explain how to rationalize the denominator of \(\frac{1}{\sqrt{3}}\).
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Multiply both the numerator and denominator by \(\sqrt{3}\) to get \(\frac{\sqrt{3}}{3}\).
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What is the result when you rationalize the denominator of \(\frac{4}{\sqrt{5}}\)?
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Multiply by \(\sqrt{5}\) to get \(\frac{4\sqrt{5}}{5}\).
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Are all surds irrational numbers?
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Yes, by definition, surds are irrational numbers because they cannot be completely resolved into a rational number.
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What is \(\sqrt{45}\) written in its simplest surd form?
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\(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\)
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Explain the difference between a surd and a rational number.
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A surd is an irrational number that cannot be expressed as a simple fraction, whereas a rational number can be expressed as a ratio of two integers.
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How do you simplify \(\sqrt{20} + \sqrt{5}\)?
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\(\sqrt{20} = 2\sqrt{5}\), so \(\sqrt{20} + \sqrt{5} = 3\sqrt{5}\).
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Why is \(8^{\frac{1}{3}}\) not a surd?
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\(8^{\frac{1}{3}}\) simplifies to 2, since it represents the cube root of 8, which is a rational number.
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