# Unit 13 Radicals

Radicals, or "Roots," are a math expression that does the opposite of what an exponent does. The most common radical is called a square root.

Example: Eight squared is sixty-four

8^{2} = 64

And: The square root of sixty-four is eight

$\sqrt{64}=8$When the radical sign in print has a superscript number in front of it, that number is called the index. If there is no index, it is assumed that the radical is number 2, or just "squared."

Radical Sign √ | ⠜ |

Termination Indicator | ⠻ |

Index Indicator | ⠣ |

In Nemeth Code, a radical expression begins with a radical sign and ends with a termination indicator. A numeric indicator is not needed directly after a radical sign because the radical sign itself is an indicator.

# Example:

$\sqrt{5}$Radical Sign | Term | Termination Ind. |

⠜ | ⠢ | ⠻ |

⠜⠢⠻

# More Examples:

$\sqrt{49}=7$⠜⠲⠔⠻⠀⠨⠅⠀⠼⠶

$\sqrt{3}+\sqrt{5}=$⠜⠒⠻⠬⠜⠢⠻⠀⠨⠅

# Radicals With an Index

When the expression is a radical other than 2, or "squared", the index of the radical must be placed in front of the radical sign. Although the printed index of the radical is usually written in superscript, in Nemeth Code, the superscript indicator is omitted and replaced with the radical’s index indicator.

# Example:

$\sqrt[3]{64}$Index Indicator | Index of Radical | Radical sign | Term | Termination Indicator |

⠣ | ⠒ | ⠜ | ⠖⠲ | ⠻ |

⠣⠒⠜⠖⠲⠻

# More Examples:

$\sqrt[5]{x}$⠣⠢⠜⠭⠻

$\sqrt[3]{15y}$⠣⠒⠜⠂⠢⠻⠽

* Notice that the effect of the radical is terminated before the "y."

${\sqrt{x}}^{2}$⠜⠭⠻⠘⠆

* Notice that the termination indicator had to be placed before the exponent.

# Radical Without a Vinculum

In print, if a radical appears without the top bar, called the vinculum, it means that the effect of the radical applies to only one term. It is the same in braille and the termination indicator is not needed.

# Examples:

√144

⠜⠂⠲⠲

√64 = 8

⠜⠖⠲⠀⠨⠅⠀⠼⠦

# More Complex Radicals

Radical expressions can contain letters, numbers, fractions, superscript or subscript, grouping symbols, baseline indicators, even other radical expressions. When this occurs, the rules for each component of the expression are followed within the radical symbols.

# Examples:

$\sqrt{{x}^{2}+{y}^{2}}$⠜⠭⠘⠆⠐⠬⠽⠘⠆⠐⠻

* Notice how the baseline indicator is used before the sign of operation and before the termination indicator.

$\sqrt{{\displaystyle \frac{x}{y}}}$⠜⠹⠭⠌⠽⠼⠻

√(a-b) + √(a+b)

⠜⠷⠁⠤⠃⠾⠬⠜⠷⠁⠬⠃⠾

$\frac{\sqrt[3]{125}}{5}$⠹⠣⠒⠜⠂⠆⠢⠻⠌⠢⠼

# Brain Boost

When a radical expression occurs within another radical expression, the inner levels of the radical expressions are nested, much like a simple fraction can be nested within a complex fraction.

Inner Radical Indicator | ⠨⠜ |

Inner Radical Termination Indicator | ⠨⠻ |

Dots 4-6 placed directly in front of the radical sign creates the first level inner radical sign and serves to nest one radical expression inside another. It is followed by an inner radical termination indicator. The entire expression is enclosed within the radical sign and termination sign of the original expression.

# Examples:

$\sqrt{a+\sqrt{b}}$⠜⠁⠬⠨⠜⠃⠨⠻⠻

$\sqrt{\sqrt[3]{1728}}$⠜⠣⠒⠨⠜⠂⠶⠆⠦⠨⠻⠻

$\sqrt[4]{12+\sqrt{{x}^{2}}}$⠣⠲⠜⠂⠆⠬⠨⠜⠭⠘⠆⠐⠨⠻⠻

* Notice the baseline indicator