D. Restrictions on Variables in Radicals
Completion requirements
D. Restrictions on Variables in Radicals
Warm Up |
Non-permissible Values for Radicals
In order for the square root of a number to be a Real Number, the radicand must be greater than or equal to zero. If given a radicand less than zero (a negative number), the square root will be part of the Complex Number system, which is not a topic of Math 20-1. Keeping a square root in the Real Number system means that the radicand will have
restrictions on the variable(s)
in a radicand containing variables. What restrictions, if any, are there for cube roots? What about \(n^{th}\) roots? Complete the Warm Up activity to explore the role of negatives with radicals.
Research Complex Numbers on the Internet to learn more about them!
\(i\)
Complete the following table:
| Radical | Value | Radical | Value |
| \(\sqrt{9}\) |
\(3\)
|
\(\sqrt[4]{81}\) | |
| \(\sqrt{0}\) |
\(0\)
|
\(\sqrt[4]{0}\) | |
| \(\sqrt{-9}\) | undefined | \(\sqrt[4]{-81}\) | |
| \(-\sqrt{9}\) |
\(-3\)
|
\(-\sqrt[4]{81}\) | |
| \(\sqrt[3]{27}\) | \(\sqrt[5]{243}\) | ||
| \(\sqrt[3]{0}\) | \(-\sqrt[5]{0}\) | ||
| \(\sqrt[3]{-27}\) | \(\sqrt[5]{-243}\) | ||
| \(-\sqrt[3]{27}\) | \(-\sqrt[5]{243}\) | ||
| \(-\sqrt[3]{-27}\) | \(-\sqrt[5]{-243}\) |
| Radical | Value | Radical | Value |
| \(\sqrt{9}\) |
\(3\)
|
\(\sqrt[4]{81}\) | \(\color{red}3\) |
| \(\sqrt{0}\) |
\(0\)
|
\(\sqrt[4]{0}\) | \(\color{red}0\) |
| \(\sqrt{-9}\) | undefined | \(\sqrt[4]{-81}\) | undefined |
| \(-\sqrt{9}\) |
\(-3\)
|
\(-\sqrt[4]{81}\) | \(\color{red}-3\) |
| \(\sqrt[3]{27}\) | \(\color{red}3\) | \(\sqrt[5]{243}\) | \(\color{red}3\) |
| \(\sqrt[3]{0}\) | \(\color{red}0\) | \(-\sqrt[5]{0}\) | \(\color{red}0\) |
| \(\sqrt[3]{-27}\) | \(\color{red}-3\) | \(\sqrt[5]{-243}\) | \(\color{red}-3\) |
| \(-\sqrt[3]{27}\) | \(\color{red}-3\) | \(-\sqrt[5]{243}\) | \(\color{red}-3\) |
| \(-\sqrt[3]{-27}\) | \(\color{red}3\) | \(-\sqrt[5]{-243}\) | \(\color{red}3\) |
In the Warm Up, you might have noticed that in order to arrive at Real Number roots, radicals with an even index cannot have negative values in the radicand. Radicals with an odd index may have negative radicands and still belong in the Real Number system.