Warm Up: Rational and Irrational Numbers
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Warm Up |
Rational and Irrational Numbers
Radicals belong to two main number sets within the Real Number System: Rational Numbers {\(Q\)}, and Irrational Numbers {\(\overline Q\)}.
Rational Numbers are numbers that can be expressed in fraction form. Some Rational Numbers are also Natural Numbers, Whole Numbers, and/or Integers.
Recall that even an Integer (negative and positive numbers including 0) can be written in fraction form.
Radicals are found in both the Rational and Irrational subsets. Complete the table below:
|
Radical
\(y = \sqrt x \) |
Value of \(y\) | \(Q\) or \(\overline Q\) |
Radical
\(y = \sqrt[3]{x}\) |
Value of \(y\) | \(Q\) or \(\overline Q\) |
|
\(\sqrt 1\)
|
\(1\) | \( \sqrt[3]{1}\) | \(1\) | \(Q\) | |
| \(\sqrt 2\) | \( \sqrt[3]{8}\) | ||||
| \(\sqrt 3\) | \(1.732...\) | \(\overline Q\) | \(\sqrt[3]{9}\) | ||
|
\(\sqrt 4\)
|
\( \sqrt[3]{27}\) | ||||
|
\(\sqrt 5\)
|
\(4\) | \(Q\) | |||
|
\(\sqrt 6\)
|
\( \sqrt[3]{70}\) | ||||
|
\(\sqrt 7\)
|
\(2.645...\) | \(\overline Q\) | \( \sqrt[3]{125}\) | \(5\) | \(Q\) |
|
\(\sqrt 8\)
|
\(\sqrt[3]{135}\) | ||||
|
\(\sqrt 9\)
|
\(6\) | \(Q\) | |||
| \(\sqrt {10}\) | \(7\) | \(Q\) |
|
Radical
\(y = \sqrt x \) |
Value of y | \(Q\) or \(\overline Q\) |
Radical
\(y = \sqrt[3]{x}\) |
Value of y | \(Q\) or \(\overline Q\) |
|
\(\sqrt 1\)
|
\(1\) | \(\color{red}Q\) | \(\sqrt[3]{1}\) | \(1\) | \(Q\) |
| \(\sqrt 2\) | \( 1.414...\) | \(\color{red}\overline Q\) | \(\sqrt[3]{8}\) |
\(2\) |
\(\color{red}Q\) |
| \(\sqrt 3\) | \(1.732...\) | \(\overline Q\) | \(\sqrt[3]{9}\) | \(2.080...\) | \(\color{red}\overline Q\) |
|
\(\sqrt 4\)
|
\( 2\) | \(\color{red}Q\) | \( \sqrt[3]{27}\) | \( 3\) | \(\color{red}Q\) |
|
\(\sqrt 5\)
|
\( 2.236...\) | \(\color{red}\overline Q\) | \(\color{red} \sqrt[3]{64}\) | \(4\) | \(Q\) |
|
\(\sqrt 6\)
|
\( 2.449...\) | \(\color{red}\overline Q\) | \( \sqrt[3]{70}\) | \( 4.121...\) | \(\color{red}\overline Q\) |
|
\(\sqrt 7\)
|
\(2.645...\) | \(\overline Q\) | \( \sqrt[3]{125}\) | \(5\) | \(Q\) |
|
\(\sqrt 8\)
|
\( 2.828...\) | \(\color{red}\overline Q\) | \( \sqrt[3]{135}\) | \( 5.129\) | \(\color{red}\overline Q\) |
|
\(\sqrt 9\)
|
\( 3\) | \(\color{red}Q\) | \(\color{red} \sqrt[3]{216}\) | \(6\) | \(Q\) |
|
\(\sqrt {10}\)
|
\(3.162...\) | \(\color{red}\overline Q\) | \(\color{red} \sqrt[3]{343}\) | \(7\) | \(Q\) |
If you need a refresher on how to use your calculator to evaluate radicals, refer to the Calculator Guide.