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• Thermodynamics
  ...ρμότης, therme, meaning "heat" and δυναμις, dynamis, meaning "power") is the study of energy conversion between heat and mechanical work, and s
  743 bytes (94 words) - 09:47, 8 July 2019
• 2010 AMC 12A Problems/Problem 23
  ...the other terms first. Then, use Fermat's Little Theorem to solve for the power of <math>2</math>.
  10 KB (1,525 words) - 09:44, 24 April 2024
• Mock AIME 1 Pre 2005 Problems/Problem 11
  ...ath> is a 12th root of unity, so both of these, when raised to the 2004-th power, become <math>3^{1002}</math>. Thus, our desired sum becomes <math>2\cdot3^
  2 KB (272 words) - 10:51, 2 July 2015
• 2010 AMC 12B Problems
  ...integer <math>n\ge2</math>, let <math>\text{pow}(n)</math> be the largest power of the largest prime that divides <math>n</math>. For example <math>\text{p
  12 KB (1,845 words) - 13:00, 19 February 2020
• Characteristic polynomial
  Exponentiating this matrix to the <math>n</math>th power will yield [[binomial coefficient]]s as follows A note about this argument: all of the power series used here are defined formally, and so we do not actually need to wo
  19 KB (3,412 words) - 14:57, 21 September 2022
• 2010 AIME I Problems/Problem 14
  ...hich <math>\lfloor \log_{10} (kn) \rfloor = 3</math>. So we go up one more power of <math>10</math> and get <math>18 + 270 + 4 = 292</math>, which is very c
  4 KB (739 words) - 22:09, 25 November 2023
• 2010 AIME I Problems/Problem 15
  ..., since they take incircles of the same size to the same circle. Also, the power of the homothety is <math>AX_{1}/AX=CX_{2}/CX</math>.
  14 KB (2,210 words) - 13:14, 11 January 2024
• Mock AIME 2 2010 Answer Key
  ...the <math>2^{5}</math> possible coefficients is multiplied by a different power of <math>x</math> because there are also <math>2^5</math> different powers ...h>E</math> be the point of tangency of the circle with <math>AC</math>. By power of a point, <math>CD \cdot CB = CE^2</math>, that is, <math>9 \cdot 25 = C^
  36 KB (6,214 words) - 20:22, 13 July 2023
• 2010 AIME II Problems/Problem 14
  ...ould be to use Power of a Point on <math>BA</math> and <math>CD</math>. By Power of a Point Theorem, <math>CP\cdot PD=1\cdot 2=BP\cdot PA</math>. Since <mat
  10 KB (1,507 words) - 00:31, 19 November 2023
• 2010 AIME II Problems/Problem 11
  ...at least once (so there is no column filled with <math>0</math>'s) and the power of each of <math>a</math>, <math>b</math>, <math>c</math> is not greater th
  6 KB (1,057 words) - 01:58, 8 January 2023
• 2010 USAJMO Problems/Problem 4
  ...ys odd. This is true because <math>2^{2n+1}</math> is even (because it's a power of <math>2</math>), so <math>2^{2n+1}+1</math> is odd, and since an odd num
  6 KB (1,001 words) - 20:21, 11 September 2021
• 1984 USAMO Problems/Problem 2
  The geometric mean of any set of <math>m</math> non-negative numbers is the <math>m</math>-th ...math> of <math>n</math> distinct positive integers such that the geometric mean of any subset of <math>S_n</math> is an integer?
  4 KB (855 words) - 03:12, 23 October 2022
• Binomial
  * A binomial to the nth power can be expanded using the [[binomial theorem]] or [[Pascal's triangle]].
  605 bytes (98 words) - 16:39, 30 June 2021
• 2010 IMO Problems/Problem 4
  [[Without loss of generality]], suppose that <math>AS > BS</math>. By Power of a Point, <math>SP^2 = SC^2 = SB \cdot SA</math>, so <math>\overline{SP}< Now consider the power of point <math>S</math> with respect to <math>\Gamma</math>.
  6 KB (895 words) - 18:32, 2 April 2021
• 2010 AMC 12B Problems/Problem 9
  ...must be a perfect <math>6</math>-th power. The smallest number whose sixth power is a multiple of <math>20</math> is <math>10</math>, because the only prime Since we have <math>2^4 \cdot 5^2 \cdot x^2</math> a power of 3, that means, by the Fundamental Theorem of Arithmetic, the exponents o
  3 KB (428 words) - 15:52, 6 May 2024
• 2010 AMC 12B Problems/Problem 25
  ...integer <math>n\ge2</math>, let <math>\text{pow}(n)</math> be the largest power of the largest prime that divides <math>n</math>. For example <math>\text{p
  4 KB (579 words) - 21:26, 16 February 2021
• 2008 iTest Problems/Problem 77
  ...Pascal triangle is due to the fact that if <math>2^k</math> is the maximum power of 2 that divides a, then <math>2^k</math> is also the maximum power of 2 that divides <math>(a-2^{k+1})</math>.
  5 KB (788 words) - 05:15, 28 January 2019
• Callebaut's Inequality
  ...th>(\frac xy-1)</math>th, <math>\frac xy</math>th, <math>\frac xy</math>th power, we get
  1 KB (236 words) - 17:30, 4 December 2010
• Python reference
  This function returned e raised to the second power.
  28 KB (4,762 words) - 21:20, 12 June 2023
• '''2.''' ''Power-mean inequality'': For <math>a_1, a_2,\cdots,a_n>0</math> and <math>k\geq l,</ma
  1 KB (197 words) - 12:12, 29 October 2016

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