Roots and 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:

8² = 64

$\sqrt{64} = 8$

The radical symbol √ is used to indicate the root. The horizontal line is called vinculum √ ̅. Everything under the vinculum is called the radicand. When the radical sign 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 2, or just "squared."

Open/Closed Radical Indicators

In braille the radicand is enclosed in radical grouping symbols.

Opening radical/root indicator	⠩
Closing radical/root indicator	⠬
Square root without vinculum	⠐⠩

A radical expression is preceded by the opening radical indicator and followed by the closing radical indicator. Since these symbols have a contracted meaning, the grade 1 indicator is needed unless the expression is already in uncontracted mode.

Example:

$\sqrt{25}$

Opening radical Indicator	Expression	Closing radical indicator
⠩	⠼⠃⠑	⠬

⠰⠩⠼⠃⠑⠬

Note: The first radical indicator requires a grade 1 indicator because the expression begins in the default mode (contracted) and has a contracted meaning (sh). The closing radical indicator does not need a grade 1 indicator because the numeric indicator has already set it in uncontracted mode.

Examples:

$2\sqrt{64}$

⠼⠃⠩⠼⠋⠙⠬

Note: No grade 1 indicators are needed because the expression begins with a numeric indicator which places it in uncontracted mode.

$- \sqrt{12} + 2$

⠐⠤⠰⠩⠼⠁⠃⠬⠐⠖⠼⠃

Notice where the vinculum ends in the next two radical expressions.

$\sqrt{x + y}$

⠰⠰⠩⠭⠐⠖⠽⠬

$\sqrt{x} + y$

⠰⠰⠩⠭⠬⠐⠖⠽

If the radical has an index other than 2, it is represented in braille as a superscript directly following the opening radical indicator.

$\sqrt[3]{5x}$

⠰⠰⠩⠔⠼⠉⠼⠑⠭⠬

$\sqrt[5]{3}y$

⠰⠰⠩⠔⠼⠑⠼⠉⠬⠽

$5\sqrt{3y^{2}}$

⠼⠑⠩⠼⠉⠽⠔⠼⠃⠬

Note: No grade 1 indicators are needed in the preceding expression because the numeric indicator sets uncontracted mode in the beginning of the expression.

When a radical appears in a subscript or superscript, the level change indicator applies to the entire item or group enclosed with the radical grouping symbols.

$x^{\sqrt{5}}$

⠰⠰⠭⠔⠩⠼⠑⠬

More Examples:

$\sqrt{a^{2} + b^{2}}$

⠰⠰⠩⠁⠔⠼⠃⠐⠖⠃⠔⠼⠃⠬

$\sqrt{\frac{4}{5}}$

⠰⠩⠼⠙⠌⠑⠬

$\frac{4}{\sqrt{32x}}$

⠰⠷⠼⠙⠨⠌⠩⠼⠉⠃⠭⠬⠾

$\frac{\sqrt{x}}{\sqrt{y}}\ = \ \sqrt{\frac{x}{y}}$

⠰⠰⠰⠷⠩⠭⠬⠨⠌⠩⠽⠬⠾⠀⠐⠶⠀⠩⠷⠭⠨⠌⠽⠾⠬⠰⠄

Note: The grade 1 passage indicator is used to span the empty spaces. Otherwise, ten grade 1 indicators would have been needed.

$\sqrt{\sqrt[4]{81} + 19}$

⠰⠰⠩⠩⠔⠼⠙⠼⠓⠁⠬⠐⠖⠼⠁⠊⠬

Radical Without a Vinculum

If a radical appears without the vinculum, the effect of the radical applies only to the term that follows it. The square root without vinculum symbol is used directly before the term and the closing radical indicator is not needed.

Examples:

√196

⠐⠩⠼⠁⠊⠋

√64 = 8

⠐⠩⠼⠋⠙⠀⠐⠶⠀⠼⠓

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