Proving one way is very simple  If n is a perfect square then  n = $a^2$  $n^7$ = $a^{14}$  $n^7$ = $a^7$($a^7$) which is obviously a perfect square for some integer a.  It's the if $n^7$ is a perfect square then n is a perfect square part that is giving me trouble. I have already tried to get hints from others and the one hint I was given was to write $n^7$ as a product of primes, but I have no clue where to go from there. I believe it can be done using modulus but I honestly have no idea.  Any hints would help thanks.
If $ a$, $ b$, $ x$, $ y$ are integers greater than $1$ such that $ a$ and $ b$ have no common factor except $1$ and $ x^a = y^b$ show that $ x = n^b$, $ y = n^a$ for some integer $ n$ greater than $1$.  I started this way: $x^a=y^b$ $=&gt;$ $x=y^{b/a}=(y^{1/a})^b$. Thus $x$ is an integer, and so is $y^{1/a}$, so $y=n^a$ for some integer $n$ greater than $1$. Then from the given relation, we get $x=n^b$.   I am not very convinced with my solution, as I feel I did some mistake. Can anyone point out where the mistake is? Any other solution is also welcome :)
Cannot translate vertice/edge will duplicate When using wish to replicate a spell, what school of magic does it count as? I'm a quarter Draconian, a quarter Dragonborn and half Half-Dragon? What are the differences between "Dragonborn", "Half-Dragon", and "Draconian" races in D&amp;D?  In the Hoard of the Dragon Queen adventure, for example there is  a Half-Dragon npc. I had assumed that when writing the adventure they were only allowed to use known material from Basic, and so renamed the Dragonborn into "Half-Dragon" for the sake of compatibility etc. However, the dragon breath rules are not at all similar for the half-dragon and the dragonborn, and I've also noticed wiki article that imply that they are different races entirely. (Thus introducing me to the term Draconian)  Are there clear differences between these three races, (in the same way there are differences between Goblins and Hobgoblins, or Dragonborn and Lizardfolk and Kobolds,) or are they really just three different setting names for the same basic race?  Answers can either be D&amp;D 5e specific, or be based on the lore from all editions of Dungeons and Dragons. (I'm assuming here that the answer will be the same regardless)  Here are two images, the first one is a "half-dragon" from the front of the Hoard of the Dragon queen module, and the second one is a "dragonborn" from the 5e player's Handbook.  I've included these images to help explain my confusion. Portrait for phone, landscape for Tablet (Android-Layout) Is there a word for a comment which makes no sense or adds nothing to the current discussion? Explicitly calling a default method in Java Spatial query to return multiple values I have a woodland scheme advice layer that has a polygon for each area of advice (brown layer in the picture). I also have a polygon layer of UK local authorities (red boundary layer in the picture). I would like to perform a spatial query which returns the local authority name for each woodland advice scheme. This would work fine if each woodland advice scheme was found in just one local authority boundary. However many are found across a boundary of two different authorities. For instance, one scheme sits across the boundaries of the authorities of Fife and Angus. The scheme is the yellow selected polygon in the picture and the red line is the authority boundary. How do I get the spatial join to return a value of Fife; Angus (or something similar to identify two values) instead of just Fife or Angus. Formula based on picklist I am trying to create a formula that will put a 1 in a hidden field if the value of a picklist is equal to a specific value.  I keep getting stuck here and I can't figure out what I am missing?  IF(ISPICKVAL( Examiner_Type__c ="Pending Examiner Status","1","",)) I recently installed Lubuntu but I have a problem with Adobe Flash Player. I always see something that looks like this when I open a video:    (source: )    The video is small, grainy, and is green and purple.  I tried to reinstall Flash and to test it in multiple browsers without success. I installed an older version of Flash but no result.  [0300]: Intel Corporation 82865G Integrated Graphics Controller [8086:2572] (rev 02)     Subsystem: Hewlett-Packard Company D530 sff(dc578av) [103c:12bc]     Kernel driver in use: i915 What is the name of this castle part? I'm trying to change the cpu frequency on my laptop (running Linux), and not having any success. Here are some details:    # uname -a Linux yoga 3.12.21-gentoo-r1 #4 SMP Thu Jul 10 17:32:31 HKT 2014 x86_64 Intel(R) Core(TM) i5-3317U CPU @ 1.70GHz GenuineIntel GNU/Linux  # cpufreq-info cpufrequtils 008: cpufreq-info (C) Dominik Brodowski 2004-2009 Report errors and bugs to cpufreq@vger.kernel.org, please. analyzing CPU 0:   driver: intel_pstate   CPUs which run at the same hardware frequency: 0   CPUs which need to have their frequency coordinated by software: 0   maximum transition latency: 0.97 ms.   hardware limits: 800 MHz - 2.60 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 800 MHz and 2.60 GHz.                   The governor "powersave" may decide which speed to use                   within this range.   current CPU frequency is 2.42 GHz (asserted by call to hardware). (similar information for cpus 1, 2 and 3)  # cat /sys/devices/system/cpu/cpu0/cpufreq/scaling_available_governors performance powersave   I initially had the userspace governor built into the kernel, but then I also tried building it as a module (with the same results); it was loaded while running the above commands (and I couldn't find any system messages when loading it):  # lsmod Module                  Size  Used by cpufreq_userspace       1525  0 (some other modules)   And here are the commands I tried for changing the frequency:  # cpufreq-set -f 800MHz Error setting new values. Common errors: - Do you have proper administration rights? (super-user?) - Is the governor you requested available and modprobed? - Trying to set an invalid policy? - Trying to set a specific frequency, but userspace governor is not available,    for example because of hardware which cannot be set to a specific frequency    or because the userspace governor isn't loaded?  # cpufreq-set -g userspace   Error setting new values. Common errors: - Do you have proper administration rights? (super-user?) - Is the governor you requested available and modprobed? - Trying to set an invalid policy? - Trying to set a specific frequency, but userspace governor is not available,    for example because of hardware which cannot be set to a specific frequency    or because the userspace governor isn't loaded?   Any ideas? How to remove the left part of a string? I have some simple python code that searches files for a string e.g. path=c:\path, where the c:\path part may vary. The current code is:  def find_path(i_file):     lines = open(i_file).readlines()     for line in lines:         if line.startswith("Path="):             return # what to do here in order to get line content after "Path=" ?   What is a simple way to get the text after Path=? I want to subtract the current frame from a previous frame, I have a image sequence and I want blender to subtract the current frame from the frame with an offset of -4. If I am on frame 5 so blender will subtract and use frame 1. Trouble Viewing Whole HDR Image as cycles world When I load an HDR image into an environment texture using Cycles Render, only a small fraction of the image appears. I have a feeling it's an easy fix, but I can not find what needs to be changed. A comparison between Grade, Mark, Score and Point Let's suppose a teacher wants to punish a student and wants to decrease his / her achieved points on a test paper; he would say something to the student; I would like to know which one of the following self-made examples may work here:        I have to take two marks of you.   I have to take two grades of you.   I have to take two scores of you.   I have to take two points of you.      For me all the first three mean the same and though the last one is not the same, but works here either. I need to get your feedback. I have been studying the pronunciation of Greek-derived words in English, and I've found an odd anomaly. There are (at least) two possible pronunciation patterns for plural word-forms that end in -&#8288;mata (which correspond to singular word-forms that end in -&#8288;ma).   Antepenult stress (expected)  It appears that it is always possible to place the stress on the antepenult (third-to-last) syllable, such as:    /ˈlɛmə/, /ˈlɛmətə/  /ˈskiːmə/, /ˈskiːmətə/  /ˌtɛrəˈtoʊmə/, /ˌtɛrəˈtoʊmətə/~/ˌtɛrəˈtɒmətə/  /ˈstɪɡmə/, /ˈstɪɡmətə/   This is what would be expected from  (I believe most or all of these words came through Latin before entering English), since the penultimate "a" in the original Greek words is short.  Maybe Latin vowel quality and stress rules are not very relevant; certainly, few English speakers know about them nowadays. And even with antepenult stress, some of these pronunciations do not seem to follow all of the traditional rules for pronouncing Latin-derived terms in English. I was reminded by some comments in this  that  in antepenult stressed syllables, we would expect tense ("long") vowels to be shortened due to  (TSL), yielding pronunciations like /ˌtɛrəˈtɒmətə/ (which I just noticed is in fact listed in the Oxford English dictionary) and ?/ˈskɛmətə/ (which is not). (Compare to genus~genera /dʒiːnəs/~/dʒɛnərə/, and perhaps opus~opera /oʊpəs/~/ɒpərə/). However, there are other Latinate plural forms that typically don't have trisyllabic laxing (matrices, also I think helices), or where both the lax- and tense-voweled pronunciations coexist (codices, apices), so I guess the apparent lack of it in schemata should not be surprising. (There are even some exceptions to TSL in singular nouns like obesity.)  Penult stress (unexpected)  The other pronunciation seems to exist only for a couple of words, in which the penultimate "a" is commonly stressed and broadened: stigma ~ stigmáta  (listed by both the Merriam-Webster dictionary and the ), and also sometimes schéma ~ schemáta /skiːˈmɑːtə/ (listed only by the ). (Neither the Merriam-Webster dictionary nor the Oxford Learner's Dictionaries list *"lemmáta").  I'm wondering why this pronunciation exists. It's an irregular plural either way, and putting the stress on the penultimate syllable creates a stress change between the singular and the plural forms.  (It's true that all of these words are rare, and some people might pronounce some of their plurals differently. The pronunciations I give here are simply the ones listed in dictionaries.)  The plural form "stigmata" is probably most common in religious or theological contexts (where it refers to the wounds of Christ), so at first I wondered if the penultimate stress was supposed to be closer to the original Greek pronunciation, but according to , the stress is on the antepenult in Greek as well.  So, what's up with this? Did the stress perhaps change in Latin over time, is this a tradition of Greek accentuation that I'm not familiar with, is this a simple matter of ignorance of the original position of the stress that has now become entrenched with time and usage, or is this somehow explainable by established English stress patterns? (For example, do English speakers have some natural preference for a penultimate stress on polysyllabic words?)  If nobody knows of a reason, I suppose I can just put it down to people encountering the word mainly in text, not knowing which syllable to stress, and choosing the penult because it sounds better to them.  (I asked this question a while back on , but nobody knew of any reason.) Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X Here X is a Hausdorff Space.  I know that singleton sets, {x}, are closed in a Hausdorff space. Although Im not sure if thats how to use the Hausdorff property.   Should I investigate $h=F(x)-x$?  Can anyone give a hint How to prove for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$? I'm new to induction so please bear with me. How can I prove using induction that, for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$?  I think $9$ can be an example since the sum of the first $9$ positive odd numbers is $1,3,5,7,9,11,13,15,17 = 81 = 9^2$, but where do I go from here. Why is chi square used when creating a confidence interval for the variance? This is a very basic question. Why do we use a chi square distribution? What is the meaning of this distribution? Why is this the distribution used for creating a confidence interval for the variance?   Every place I google for an explanation just presents this a fact, explaining when to use chi, but not explaining why to use chi, and why it look the way it does.  Many thanks to anyone that can point me towards the right direction and that is - really understanding why I am using chi when I am creating a confidence interval for the variance. Is a strong background in maths a total requisite for ML? I'm starting to want to advance my own skillset and I've always been fascinated by machine learning. However, six years ago instead of pursuing this I decided to take a completely unrelated degree to computer science.   I have been developing software and applications for about 8-10 years now, so I have a good handle but I just can't seem to penetrate the maths side of machine learning/probabilities/statistics.   I start looking at learning material and on the first page it might include something which confuses me and immediately sets up a barrier in my learning.   Is a strong background in maths a total requisite for ML? Should I try and fill in the blanks of my maths before continuing with ML? Can self learning really work for just a developer without any hard computer science background?    Related question: