Template:Nroot

The following documentation is located at Template:Nroot/doc. [<Edit> Template:Nroot/doc]
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The {{nroot}} mathematical formatting template and/or mathematical function template typesets either finite or infinite [generalized] nested radicals in either HTML+CSS or LaTeX, or approximates (up to 14 decimal digits) the numerical result.

Usage

For finite nested radicals, use either

{{nroot|r0;; m1//r1,, m2//r2,, ...,, mk//rk|format}}

or

{{nroot|r0;; m1//r1//i1,, m2//r2//i2,, ...,, mk//rk//ik|format}}

while for infinite nested radicals, use either (where in either case the last rk is left blank)

{{nroot|r0;; m1//r1,, m2//r2,, ...,, mk//|format}}

or

{{nroot|r0;; m1//r1//i1,, m2//r2//i2,, ...,, mk// //ik|format}}

where

• r0 followed by two consecutive semi-columns is the integer part;
• mj, with 1   ≤   j   ≤   k   ≤   8, are the “partial multipliers” (only the first eight “partial multipliers” are considered, the others are ignored);
• rj, with 1   ≤   j   ≤   k   ≤   8, are the “partial radicands” (only the first eight “partial radicands” are considered, the others are ignored);
• ij, with 1   ≤   j   ≤   k   ≤   8, are the “partial indices” (only the first eight “partial indices” are considered, the others are ignored);

and where format is from (currently, lowercase or uppercase gives the same result)

• htm: HTML+CSS,
• HTM: HTML+CSS,
• tex: LaTeX,
• TEX: LaTeX, or
• #: numerical result (up to 14 decimal digits) of nested radical (up to the last provided rk, k   ≤   8).

Notes:

• If the last rk is empty, the nested radicals are considered infinite, otherwise considered finite;
• “partial multipliers” equal to 1 are automatically blanked (append a null space, e.g. 1{{sp|0}} to force 1 to appear);
• “partial radicands” equal to 0 (as well as the + sign that would have followed) are automatically blanked (append a null space, e.g. 0{{sp|0}} to force 0 followed by a + sign to appear).

Examples

The code

: {{math|? {{=|sp}} {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|HTM}} {{=|sp}} {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|#}}{{...|ldots}}
|tex = ? = {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|TEX}} = {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|#}}\ldots  
|&&}}

yields the display style HTML+CSS (Check 14 digits: 0 + 2 * sqrt(1 + 4 * sqrt(2 + 6 * sqrt(3 + 8 * sqrt(4 + 10 * sqrt(5 + 12 * sqrt(6 + 14 * sqrt(7 + 16 * sqrt(8)))))))) = 11.258138412482039...)

?  =  2  2√  1 + 4  2√  2 + 6  2√  3 + 8  2√  4 + 10  2√  5 + 12  2√  6 + 14  2√  7 + 16  2√  8  =  11.258138412482…

The code

: {{math|? {{=|sp}} {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|HTM}} {{=|sp}} {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|#}}{{...|ldots}}
|tex = ? = {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|TEX}} = {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|#}}\ldots  
|$$}}

yields the display style LaTeX (Check 14 digits: 0 + 2 * sqrt(1 + 4 * sqrt(2 + 6 * sqrt(3 + 8 * sqrt(4 + 10 * sqrt(5 + 12 * sqrt(6 + 14 * sqrt(7 + 16 * sqrt(8)))))))) = 11.258138412482039...)

${\begin{array}{l}\displaystyle {?={2{\sqrt {1\,+\,4{\sqrt {2\,+\,6{\sqrt {3\,+\,8{\sqrt {4\,+\,10{\sqrt {5\,+\,12{\sqrt {6\,+\,14{\sqrt {7\,+\,16{\sqrt {8}}}}}}}}}}}}}}}}}=11.258138412482\ldots }\end{array}}$

The code

: {{math|? {{=|sp}} {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//|HTM}}
|tex = ? = {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//|TEX}}
|&&}}

yields the display style HTML+CSS

?  =  2  2√  1 + 4  2√  2 + 6  2√  3 + 8  2√  4 + 10  2√  5 + 12  2√  6 + 14  2√  7 + 16   2√  ⋯

The code

: {{math|? {{=|sp}} {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//|HTM}}
|tex = ? = {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//|TEX}}
|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {?={2{\sqrt {1\,+\,4{\sqrt {2\,+\,6{\sqrt {3\,+\,8{\sqrt {4\,+\,10{\sqrt {5\,+\,12{\sqrt {6\,+\,14{\sqrt {7\,+\,16{\sqrt {\cdots }}}}}}}}}}}}}}}}}}\end{array}}$

The code

: {{math|? {{=|sp}} {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4|HTM}}
|tex = ? = {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4|TEX}} 
|&&}}

yields the display style HTML+CSS

?  =  2  2√  1 + 4  2√  2 + 6  2√  3 + 8  2√  4

The code

: {{math|? {{=|sp}} {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4|HTM}}
|tex = ? = {{nroot|0;; 2//1,, 4//2,, 6//3,, 8//4|TEX}} 
|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {?={2{\sqrt {1\,+\,4{\sqrt {2\,+\,6{\sqrt {3\,+\,8{\sqrt {4}}}}}}}}}}\end{array}}$

The code

: {{math|? {{=|sp}} {{nroot|;; 2//1,, 4//2,, 6//3,, 8//|HTM}}
|tex = ? = {{nroot|;; 2//1,, 4//2,, 6//3,, 8//|TEX}} 
|&&}}

yields the display style HTML+CSS

?  =  2  2√  1 + 4  2√  2 + 6  2√  3 + 8   2√  ⋯

The code

: {{math|? {{=|sp}} {{nroot|;; 2//1,, 4//2,, 6//3,, 8//|HTM}}
|tex = ? = {{nroot|;; 2//1,, 4//2,, 6//3,, 8//|TEX}} 
|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {?={2{\sqrt {1\,+\,4{\sqrt {2\,+\,6{\sqrt {3\,+\,8{\sqrt {\cdots }}}}}}}}}}\end{array}}$

The code

: {{math|? {{=|sp}} {{nroot|0;; ''x''//1,, ''x''{{^|2}}//2,, ''x''{{^|3}}//3,, ''x''{{^|4}}//4,, ''x''{{^|5}}//5,, <!--
-->''x''{{^|6}}//6,, ''x''{{^|7}}//7,, ''x''{{^|8}}//|HTM}}
|tex = ? = {{nroot|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//4,, ''x''{{^|5|tex}}//5,, <!--
-->''x''{{^|6|tex}}//6,, ''x''{{^|7|tex}}//7,, ''x''{{^|8|tex}}//|TEX}}
|&&}}

yields the display style HTML+CSS

?  =  x  2√  1 + x 2  2√  2 + x 3  2√  3 + x 4  2√  4 + x 5  2√  5 + x 6  2√  6 + x 7  2√  7 + x 8   2√  ⋯

The code

: {{math|? {{=|sp}} {{nroot|0;; ''x''//1,, ''x''{{^|2}}//2,, ''x''{{^|3}}//3,, ''x''{{^|4}}//4,, ''x''{{^|5}}//5,, <!--
-->''x''{{^|6}}//6,, ''x''{{^|7}}//7,, ''x''{{^|8}}//|HTM}}
|tex = ? = {{nroot|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//4,, ''x''{{^|5|tex}}//5,, <!--
-->''x''{{^|6|tex}}//6,, ''x''{{^|7|tex}}//7,, ''x''{{^|8|tex}}//|TEX}}
|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {?={x{\sqrt {1\,+\,x^{2}{\sqrt {2\,+\,x^{3}{\sqrt {3\,+\,x^{4}{\sqrt {4\,+\,x^{5}{\sqrt {5\,+\,x^{6}{\sqrt {6\,+\,x^{7}{\sqrt {7\,+\,x^{8}{\sqrt {\cdots }}}}}}}}}}}}}}}}}}\end{array}}$

The code

: {{math|? {{=|sp}} {{nroot|0;; ''x''//1,, ''x''{{^|2}}//2,, ''x''{{^|3}}//3,, ''x''{{^|4}}//|HTM}}
|tex = ? = {{nroot|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//|TEX}}
|&&}}

yields the display style HTML+CSS

?  =  x  2√  1 + x 2  2√  2 + x 3  2√  3 + x 4   2√  ⋯

The code

: {{math|? {{=|sp}} {{nroot|0;; ''x''//1,, ''x''{{^|2}}//2,, ''x''{{^|3}}//3,, ''x''{{^|4}}//|HTM}}
|tex = ? = {{nroot|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//|TEX}}
|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {?={x{\sqrt {1\,+\,x^{2}{\sqrt {2\,+\,x^{3}{\sqrt {3\,+\,x^{4}{\sqrt {\cdots }}}}}}}}}}\end{array}}$

Examples with root indices

The code

: {{math|? {{=|sp}} {{nroot|{{op|-}}1;; 2//1//2,, 4//2//3,, 6//3//4,, 8//4//5,, 10//5//6,, 12//6//7,, 14//7//8,, 16// //9|HTM}} {{=|sp}} <!--
-->{{nroot|{{op|-}}1;; 2//1//2,, 4//2//3,, 6//3//4,, 8//4//5,, 10//5//6,, 12//6//7,, 14//7//8,, 16// //9|#}}{{...|ldots}}
|tex = ? = {{nroot|{{op|-}}1;; 2//1//2,, 4//2//3,, 6//3//4,, 8//4//5,, 10//5//6,, 12//6//7,, 14//7//8,, 16// //9|TEX}} = <!--
-->{{nroot|{{op|-}}1;; 2//1//2,, 4//2//3,, 6//3//4,, 8//4//5,, 10//5//6,, 12//6//7,, 14//7//8,, 16// //9|#}}\ldots
|&&}}

yields the display style HTML+CSS (Check 14 digits: -1 + 2 * ( 1 + 4 * ( 2 + 6 * ( 3 + 8 * ( 4 + 10 * ( 5 + 12 * ( 6 + 14 * ( 7 )^(1/8))^(1/7))^(1/6))^(1/5))^(1/4))^(1/3))^(1/2) = 5.5454104248858983...)

?  =  −1 + 2  2√  1 + 4  3√  2 + 6  4√  3 + 8  5√  4 + 10  6√  5 + 12  7√  6 + 14  8√  7 + 16   9√  ⋯   =  5.5454104248859…

The code

: {{math|? {{=|sp}} {{nroot|{{op|-}}1;; 2//1//2,, 4//2//3,, 6//3//4,, 8//4//5,, 10//5//6,, 12//6//7,, 14//7//8,, 16// //9|HTM}} {{=|sp}} <!--
-->{{nroot|{{op|-}}1;; 2//1//2,, 4//2//3,, 6//3//4,, 8//4//5,, 10//5//6,, 12//6//7,, 14//7//8,, 16// //9|#}}{{...|ldots}}
|tex = ? = {{nroot|{{op|-}}1;; 2//1//2,, 4//2//3,, 6//3//4,, 8//4//5,, 10//5//6,, 12//6//7,, 14//7//8,, 16// //9|TEX}} = <!--
-->{{nroot|{{op|-}}1;; 2//1//2,, 4//2//3,, 6//3//4,, 8//4//5,, 10//5//6,, 12//6//7,, 14//7//8,, 16// //9|#}}\ldots
|$$}}

yields the display style LaTeX (Check 14 digits: -1 + 2 * ( 1 + 4 * ( 2 + 6 * ( 3 + 8 * ( 4 + 10 * ( 5 + 12 * ( 6 + 14 * ( 7 )^(1/8))^(1/7))^(1/6))^(1/5))^(1/4))^(1/3))^(1/2) = 5.5454104248858983...)

${\begin{array}{l}\displaystyle {?={-1\,+\,2{\sqrt[{2}]{1\,+\,4{\sqrt[{3}]{2\,+\,6{\sqrt[{4}]{3\,+\,8{\sqrt[{5}]{4\,+\,10{\sqrt[{6}]{5\,+\,12{\sqrt[{7}]{6\,+\,14{\sqrt[{8}]{7\,+\,16{\sqrt[{9}]{\cdots }}}}}}}}}}}}}}}}}=5.5454104248859\ldots }\end{array}}$

Mathematical examples

See: mathematical examples.

See also

• {{cfrac}} (typesets either finite or infinite [generalized] continued fractions in either HTML+CSS or LaTeX) (particular case of finite or infinite [generalized] nested radicals with root indices all equal to  − 1)