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# Lesson 5.2: Exponents (Superscripts)

## Symbols

$\text{level change up indicator (exponent or superscript)}$

$\text{braille grouping, opening}$

$\text{braille grouping, closing}$

## Explanation

A quantity displayed above the normal line of text and generally in a smaller point size is known as a superscript. Superscripts are commonly used to indicate an exponent or raising to a power. Exponents are just one example of when superscripts are used in mathematics. Superscripts can appear to the right or left of a symbol or group of symbols, however they do not appear directly over or under them.

In braille, the level change up indicator is used to identify the item that follows as being in the superscript position. The extent of the level change up indicator is the next item. An item is defined as any one of the following groupings if appearing immediately after the level change up indicator:

• An entire number, i.e. the initiating numeric symbol and all succeeding symbols within the numeric mode thus established (which would include any interior decimal points, commas, separator spaces, or simple numeric fraction lines).
• An entire general fraction, enclosed in fraction indicators.
• An arrow.
• An arbitrary shape.
• Any expression enclosed in matching pairs of round parentheses, square brackets or curly braces.
• Any expression enclosed in the braille grouping indicators.
• If none of the foregoing apply, the item is simply the next individual symbol.

The braille level change up indicator is dots three five. The opening braille grouping symbol is dots one two six and the closing braille grouping symbol is dots three four five. All three symbols also have a grade 2 meaning, therefore the symbols must be in grade 1 mode. Grade 1 mode is set by the numeric indicator or by grade 1 indicators.

The level change up indicator terminates numeric mode. The numeric indicator must be used with a numeral that appears in the superscript position.

### Example 1

${9}^{2}$
⠼⠊⠔⠼⠃

### Example 2

${3}^{2}×{4}^{3}$
⠼⠉⠔⠼⠃⠐⠦⠼⠙⠔⠼⠉

### Example 3

$\text{What is the value of}\phantom{\rule{.3em}{0ex}}{2}^{6}\text{?}$
⠠⠱⠁⠞⠀⠊⠎⠀⠮⠀⠧⠁⠇⠥⠑⠀⠷⠀⠼⠃⠔⠼⠋⠦

### Example 4

$\left(6×{5}^{2}\right)$
⠐⠣⠼⠋⠐⠦⠼⠑⠔⠼⠃⠐⠜

### Example 5

${2}^{x}$
⠼⠃⠔⠭

The grade 1 indicator is required when the superscript indicator is used in a grade 2 context.

### Example 6

${y}^{x}$
⠽⠰⠔⠭

### Example 7

${n}^{x}+{y}^{x}$
⠰⠰⠝⠔⠭⠐⠖⠽⠔⠭

In Example 8, a grade 1 symbol indicator is used with the first level change up indicator. The numeric indicator turns on grade 1 mode for the second level change up indicator.

### Example 8

${n}^{3}-2{n}^{2}$
⠝⠰⠔⠼⠉⠐⠤⠼⠃⠝⠔⠼⠃

Braille grouping indicators must be used when the quantity in the superscript position consists of more than one item. The symbols must be in grade 1 mode since they also have an alternate grade 2 meaning. In Examples 9 and 10, Grade 1 mode is turned on by the numeric indicator and remains in effect for the remainder of the sequence.

### Example 9

${2}^{b}+4$
⠼⠃⠔⠃⠐⠖⠼⠙

### Example 10

${2}^{b+4}$
⠼⠃⠔⠣⠃⠐⠖⠼⠙⠜

### Example 11

${2}^{12}$
⠼⠃⠔⠼⠁⠃

In Example 12, a grade 1 word indicator is used to place the entire expression in grade 1 mode. This avoids interrupting the mathematical expression with multiple grade 1 indicators.

### Example 12

${y}^{2x}$
⠰⠰⠽⠔⠣⠼⠃⠭⠜

Sometimes in print the quantity in the superscript position is written to the left of the baseline symbol instead of to the right. In this case, the index expression should immediately precede the baseline symbol. A grade 1 indicator is required unless grade 1 mode is already in effect.

### Example 13

${}^{i}a$
⠰⠔⠊⠁

Superscripts can have multiple levels. Braille grouping indicators must be used when the quantity in the superscript position consists of more than one “item.” A superscript of a superscript requires the level change up indicator followed by the superscript material. This combination of symbols is considered more than one item therefore it must be written within braille grouping indicators to be read as a single item.

### Example 14

${7}^{{x}^{2}}$
⠼⠛⠔⠣⠭⠔⠼⠃⠜

### Example 15

${7}^{{x}^{2}}+4$
⠼⠛⠔⠣⠭⠔⠼⠃⠜⠐⠖⠼⠙

### Example 16

${7}^{{x}^{2}+4}$
⠼⠛⠔⠣⠭⠔⠼⠃⠐⠖⠼⠙⠜