Understanding Surds

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What is a surd?

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.

Give an example of a surd.

The square root of 2 (\(\sqrt{2}\)) is a surd because it cannot be simplified to a rational number.

Is \(\sqrt{4}\) a surd? Why or why not?

No, \(\sqrt{4}\) is not a surd because it simplifies to 2, a rational number.

Simplify \(\sqrt{50}\) into its simplest surd form.

\(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\)

How do you multiply surds?

To multiply surds, multiply the numbers under the roots and then simplify if possible. For example, \(\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6\).

Can surds be added directly like integers? Why?

No, surds can only be added or subtracted if they have the same radicand, similar to adding like terms in algebra.

What is the product of \(\sqrt{6} \times \sqrt{6}\)?

The product is 6, because \(\sqrt{6} \times \sqrt{6} = (\sqrt{6})^2 = 6\).

What is the simplest form of \(\sqrt{8} + 2\sqrt{2}\)?

\(\sqrt{8} = 2\sqrt{2}\), so \(\sqrt{8} + 2\sqrt{2} = 4\sqrt{2}\).

Explain how to rationalize the denominator of \(\frac{1}{\sqrt{3}}\).

Multiply both the numerator and denominator by \(\sqrt{3}\) to get \(\frac{\sqrt{3}}{3}\).

What is the result when you rationalize the denominator of \(\frac{4}{\sqrt{5}}\)?

Multiply by \(\sqrt{5}\) to get \(\frac{4\sqrt{5}}{5}\).

Are all surds irrational numbers?

Yes, by definition, surds are irrational numbers because they cannot be completely resolved into a rational number.

What is \(\sqrt{45}\) written in its simplest surd form?

\(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\)

Explain the difference between a surd and a rational number.

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.

How do you simplify \(\sqrt{20} + \sqrt{5}\)?

\(\sqrt{20} = 2\sqrt{5}\), so \(\sqrt{20} + \sqrt{5} = 3\sqrt{5}\).

Why is \(8^{\frac{1}{3}}\) not a surd?

\(8^{\frac{1}{3}}\) simplifies to 2, since it represents the cube root of 8, which is a rational number.