# Lesson 11.6: More Complex Radicals

The radical sign, dots three four five, and the termination indicator, dots one two four five six, were already introduced as they are used to show square roots.

The level of a square root is two. Since it is the most common of the roots, the level, or index, of the root is not shown. It is commonly understood to be two. Radicals of a different index must be designated by a numeral to indicate their level. Third level radicals are referred to as the cube root; radicals of higher levels are referred to by their ordinal names, such as fourth or fifth root.

## Indicating roots of other levels

In print, to indicate that a radical shows the third root, a small numeral three, referred to as the index of the radical, is placed in the crook of the radical sign. The number from which the root is to be extracted, the radicand, is enclosed under a horizontal bar, referred to as the vinculum.

In braille, the index-of-radical indicator, dots one two six, precedes the numeral which indicates the level of the radical, index of the radical. The index of the radical is followed by the radical sign, dots three four five, the radicand, and the termination indicator, dots one two four five six. The index-of-radical sign and the termination sign are braille indicators which have no corresponding symbols in print.

Treat the entire root, from its index-of-the-radical indicator to its termination indicator, as if it were a single value or symbol. Apply the general rules governing all other braille symbols as previously described.

## Radicals and Fractions

A radical sign may encompass a fraction or a fraction may contain radicals in its numerator, denominator, or both. A radical may also be shown with fractional exponents instead of being displayed with the radical sign. Since the radical sign and the termination indicator are signs of grouping, and since the fraction line, opening fraction indicator, and closing fraction indicator are also signs of grouping, it is important to know what portion of a problem is being encompassed by another.

## Nested Radicals

A nested radical is composed of two or more radicals where one or more is embedded in another. In print, several nested vinculums are displayed successively above each other. In braille, nested radicals are brailled in a horizontal manner. The depth of the inner radical is shown with the order-of-radical indicator, dots four six. This indicator must precede the index-of-radical indicator, if one is present; the inner radical symbol; and the termination indicator for the inner radical. For every opening radical indicator, there is a matching closing termination indicator. For every opening order-of-radical indicator, there is a matching closing order-of-radical indicator.

It is the quantity of order-of-radical indicators that signifies the depth or level of each of the radicals. Order-of-radical indicators also appear before termination indicators to denote which radical has ended. The outermost radical has no depth and therefore is not required to have an order-of-radical indicator.

## Exponents that appear within radicals

When exponents appear within radicals, the baseline indicator is used to indicate a return to the baseline. Since termination indicators are at the same level as radical signs, baseline indicators are required before closing order-of-radical termination indicators.

### Example 1

$\sqrt[3]{64}$

⠣⠒⠜⠖⠲⠻

### Example 2

$\sqrt[3]{64}=4$

⠣⠒⠜⠖⠲⠻⠀⠨⠅⠀⠼⠲

### Example 3

$3+\sqrt[3]{64}$

⠼⠒⠬⠣⠒⠜⠖⠲⠻

### Example 4

$\sqrt[\mathrm{n}]{64}$

⠣⠝⠜⠖⠲⠻

### Example 5

$5\sqrt[3]{64}$

⠼⠢⠣⠒⠜⠖⠲⠻

$5\left(\sqrt[3]{64}\right)$

⠼⠢⠷⠣⠒⠜⠖⠲⠻⠾

### Example 6

$\sqrt[3]{\frac{1}{8}}$

⠣⠒⠜⠹⠂⠌⠦⠼⠻

### Example 7

$\sqrt{\sqrt[3]{8000}}$

⠜⠨⠣⠒⠜⠦⠴⠴⠴⠨⠻⠻

### Example 8

$\sqrt{\sqrt{64}+2}$

⠜⠨⠜⠖⠲⠨⠻⠬⠆⠻

### Example 9

$\sqrt{{x}^{4}}$

⠜⠭⠘⠲⠐⠻