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- ...root-mean power]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> th ...~~-1<n_3<0,~~n_4<-1</math>, and <math>n_i</math> is the <math>\textbf{root mean power}</math>.5 KB (913 words) - 15:44, 14 August 2024
- ...<math>y</math> is a square root of <math>x</math> if <math>x</math> is the square of <math>y</math>. ...nsider only [[positive number|positive]] [[real number|reals]], the square root [[function]] is the [[Function/Introduction#The_Inverse_of_a_Function|inver2 KB (291 words) - 22:14, 4 February 2025
- ...qrt{\frac{x^2_1+x^2_2+\dots+x^2_n}{n}}</math>. This is the second [[power mean]] of the <math>x_i</math>. It is so-named because it is the square root of the mean of the squares of the <math>x_i</math>.543 bytes (86 words) - 13:41, 8 May 2013
- #REDIRECT [[Root-mean-square]]30 bytes (2 words) - 20:05, 16 July 2007
- #REDIRECT [[Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality]]86 bytes (7 words) - 21:26, 20 December 2007
- #REDIRECT [[Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality]]86 bytes (7 words) - 21:54, 20 December 2007
- The '''square root property''' is a [[method]] for solving [[quadratic equation|quadratic equa take the square root of both sides626 bytes (98 words) - 23:55, 15 May 2014
- ==== Proof that any nonperfect square positive integer is irrational ==== ...no perfect square times a nonperfect square positive integer is a perfect square. Therefore <math>\sqrt{n}</math> is irrational.435 bytes (72 words) - 14:17, 19 June 2019
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- ...asy. In fact, all [[quadratic equation]]s can be solved by completing the square. ...part. If <math>9</math> was added to this, then we would have a [[perfect square]], <math>(x-3)^2=x^2-6x+9</math>. To do this, add <math>7</math> to each s2 KB (422 words) - 16:20, 5 March 2023
- == Square root simplification/modification == ...late the area of a certain triangle with side lengths with quantities with square roots. One can remember this formula by noticing that when finding the cosi4 KB (685 words) - 20:11, 24 February 2025
- ====What is a root?==== ...variable that will make the polynomial equal zero. For an example, 2 is a root of <math>x^2 - 4</math> because <math>2^2 - 4 = 0</math>. For some polynom6 KB (1,100 words) - 15:57, 30 August 2024
- ...er methods, such as [[completing the square]], [[factoring]], and [[square root property]] do not work or are too tedious. Add <math>\frac{b^{2}}{4a^{2}}</math> to both sides in order to complete the square:2 KB (281 words) - 10:11, 2 February 2025
- ===Proof that the square root of 2 is irrational===2 KB (374 words) - 14:01, 21 August 2022
- ...</math>, then <math>(a+1)^p \equiv a+1 \pmod{p}</math>, as desired. <math>\square</math> ...text{natural numbers relatively prime to and less than}\ n\}</math> <math>\square</math>15 KB (2,618 words) - 12:03, 19 February 2025
- ...c) <math>A=2</math>, where only non-negative real numbers are admitted for square roots? The square roots imply that <math>x\ge \frac{1}{2}</math>.4 KB (612 words) - 00:19, 7 March 2025
- ...The range is the set of all non-negative real numbers, because the square root can never return a negative value.10 KB (1,761 words) - 03:16, 12 May 2023
- ...t just circling all remaining numbers once a number exceeding the [[square root]] of the maximal number we are interested in is circled (in our example it1 KB (212 words) - 21:16, 7 December 2007
- ...be rewritten in the decimal system (base-10) before evaluating the square root. To do this, multiply and add <math>6\cdot 8^1+1\cdot 8^0=48+1=49. \sqrt{44 KB (547 words) - 17:23, 30 December 2020
- The plus or minus does not mean that there are two answers, but that the sign of the expression depends on ...te <math>\frac{1}{2} x</math> instead of <math>x</math>. Taking the square root then yields the desired half-angle identities for sine and cosine. As for t8 KB (1,397 words) - 21:55, 20 January 2024
- ...root-mean power]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> th ...~~-1<n_3<0,~~n_4<-1</math>, and <math>n_i</math> is the <math>\textbf{root mean power}</math>.5 KB (913 words) - 15:44, 14 August 2024
- An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The fi The sum of the first <math>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</mat959 bytes (156 words) - 14:22, 20 January 2025
- ...+ 2 \sqrt{FC^2 + 64} = 8FC.</cmath> Grouping and getting rid of the square root gives us <math>FC^2 + 64 = 9FC^2</math>, meaning that <math>FC = 2 \sqrt{2}5 KB (811 words) - 11:44, 30 November 2024
- ...e [[positive]] [[integer]]s and <math> r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> ...<math>200 \pm 50\sqrt{7}</math>. Since <math>y < x</math>, use the smaller root, <math>200 - 50\sqrt{7}</math>.13 KB (2,080 words) - 13:14, 23 July 2024
- ...<math>y</math> is a square root of <math>x</math> if <math>x</math> is the square of <math>y</math>. ...nsider only [[positive number|positive]] [[real number|reals]], the square root [[function]] is the [[Function/Introduction#The_Inverse_of_a_Function|inver2 KB (291 words) - 22:14, 4 February 2025
- ...with [[diameter]] <math> d </math> is contained in a [[square (geometry) | square]] whose sides have length 8. Given the maximum value of <math> d </math> is ...arranged so that the center of the semicircle lies on one diagonal of the square.4 KB (707 words) - 11:11, 16 September 2021
- ...triangle ADC</math> (where we square everything to avoid taking the square root) we see14 KB (2,340 words) - 16:38, 21 August 2024
- ...their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? ...>p</math> are relatively prime, and <math>n</math> is not divisible by the square of any prime, find <math>m + n + p.</math>7 KB (1,127 words) - 09:02, 11 July 2023
- Taking the square root of the equation <math>x^{24}=w</math> results in <math>x^{12}=w^{\frac{1}{2 ...two equations, we get, <math>x^{24}y^{40}= w^{2}</math>. Taking the square root of this, we find that <math>x^{12}y^{20}=w</math>. Recall, <math>x^{12}y^{14 KB (641 words) - 10:19, 6 January 2025
