{"id":2174,"date":"2026-06-18T18:27:29","date_gmt":"2026-06-18T16:27:29","guid":{"rendered":"https:\/\/okoster.hu\/tudaster\/?p=2174"},"modified":"2026-06-18T19:12:50","modified_gmt":"2026-06-18T17:12:50","slug":"emeletes-gyok-avagy-a-matematika-orak-remalma","status":"publish","type":"post","link":"https:\/\/okoster.hu\/tudaster\/2026\/06\/18\/emeletes-gyok-avagy-a-matematika-orak-remalma\/","title":{"rendered":"Emeletes gy\u00f6k, avagy a matematika \u00f3r\u00e1k r\u00e9m\u00e1lma"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Ezt a feladatot, a Facebookon <a href=\"https:\/\/www.facebook.com\/matematica101\/reels\/\">>> Matem\u00e1tica Simples digit\u00e1lis tartalomk\u00e9sz\u00edt\u0151 &lt;&lt;<\/a> tette k\u00f6zz\u00e9 egy <a href=\"https:\/\/www.facebook.com\/reel\/1684035009542254\">>> Reels vide\u00f3ban &lt;&lt;<\/a>.<\/p>\n\n\n\n<div style=\"height:15px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div style=\"height:15px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mtable columnalign=\"left left\"><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><msqrt><mrow><mn>9<\/mn><msqrt><mrow><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><\/mrow><\/msqrt><\/mrow><\/msqrt><mo>=<\/mo><mo form=\"postfix\" stretchy=\"false\">?<\/mo><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><mtext>a.)<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mn>3<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><mtext>b.)<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mn>3<\/mn><msqrt><mn>3<\/mn><\/msqrt><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><mtext>c.)<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><msup><mn>3<\/mn><mrow><mn>3<\/mn><mi>\/<\/mi><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><mtext>d.)<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mn>9<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{array}{ll}\n\\sqrt{9\\sqrt{9\\sqrt{9}}} = ? &amp; \\qquad \\text{a.)} \\ \\ \\ \\ 3 \\\\\n                             &amp; \\qquad \\text{b.)} \\ \\ \\ \\ 3\\sqrt{3} \\\\\n                             &amp; \\qquad \\text{c.)} \\ \\ \\ \\ 3^{3\/2} \\\\\n&amp; \\qquad \\text{d.)} \\ \\ \\ \\ 9\n\\end{array}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div style=\"height:15px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div style=\"height:15px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">A megold\u00e1st a &#8222;lusta matekos&#8221; metodik\u00e1j\u00e1val k\u00e9sz\u00edtettem el. A var\u00e1zslat, a feladat egyszer\u0171 megold\u00e1sa abban rejlik, hogy bevezet\u00fcnk egy &#8216;a&#8217;-val jelzett v\u00e1ltoz\u00f3t. Ez seg\u00edt abban, hogy a megold\u00e1shoz nagyon minim\u00e1lis sz\u00e1mol\u00f3g\u00e9p haszn\u00e1lat sz\u00fcks\u00e9ges. \u00c9n kb. k\u00e9tszer vettem el\u0151 a sz\u00e1mol\u00f3g\u00e9pemet.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>De nem h\u00fazom tov\u00e1bb a sz\u00f3t, l\u00e1ssuk a megold\u00e1st:<\/strong><\/p>\n\n\n\n<div style=\"height:15px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div style=\"height:15px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mtable columnalign=\"left left\"><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><msqrt><mn>9<\/mn><\/msqrt><mo>=<\/mo><mn>3<\/mn><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><mtext>\ud835\udc08\ud835\udc1f<\/mtext><mspace width=\"1em\"><\/mspace><mrow><msqrt><mrow><mn>\ud835\udfd7<\/mn><msqrt><mrow><mn>\ud835\udfd7<\/mn><msqrt><mn>\ud835\udfd7<\/mn><\/msqrt><\/mrow><\/msqrt><\/mrow><\/msqrt><mo>=<\/mo><mroot><mi>\ud835\udc1a<\/mi><mn>\ud835\udfd2<\/mn><\/mroot><\/mrow><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><msqrt><mrow><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mrow><mn>9<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mn>27<\/mn><\/msqrt><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><mi>a<\/mi><mo>=<\/mo><msup><mrow><mo fence=\"true\" form=\"prefix\">{<\/mo><msqrt><mrow><mn>9<\/mn><msqrt><mrow><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><\/mrow><\/msqrt><\/mrow><\/msqrt><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mn>4<\/mn><\/msup><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><msqrt><mrow><mn>9<\/mn><msqrt><mrow><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><\/mrow><\/msqrt><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mrow><mn>9<\/mn><msqrt><mn>27<\/mn><\/msqrt><\/mrow><\/msqrt><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><msup><mrow><mo fence=\"true\" form=\"prefix\">{<\/mo><msqrt><mrow><mn>9<\/mn><msqrt><mrow><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><\/mrow><\/msqrt><\/mrow><\/msqrt><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><mn>9<\/mn><msqrt><mrow><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><\/mrow><\/msqrt><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><msup><mrow><mo fence=\"true\" form=\"prefix\">{<\/mo><mn>9<\/mn><msqrt><mrow><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><\/mrow><\/msqrt><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mn>9<\/mn><mn>2<\/mn><\/msup><mo>\u00d7<\/mo><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><mo>=<\/mo><mn>81<\/mn><mo>\u00d7<\/mo><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><mo>=<\/mo><mn>729<\/mn><msqrt><mn>9<\/mn><\/msqrt><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><mi>a<\/mi><mo>=<\/mo><mn>729<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><mo>=<\/mo><mn>2187<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><msqrt><mrow><mn>9<\/mn><msqrt><mrow><mn>9<\/mn><msqrt><mn>9<\/mn><\/msqrt><\/mrow><\/msqrt><\/mrow><\/msqrt><mo>=<\/mo><mroot><mn>2187<\/mn><mn>4<\/mn><\/mroot><mo>\u21a6<\/mo><mn>2187<\/mn><mo>=<\/mo><msup><mn>3<\/mn><mn>7<\/mn><\/msup><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0pt;padding-right:5.9776pt;\"><mrow><mroot><mn>2187<\/mn><mn>4<\/mn><\/mroot><mo>=<\/mo><mroot><msup><mn>3<\/mn><mn>7<\/mn><\/msup><mn>4<\/mn><\/mroot><mo>=<\/mo><msup><mn>3<\/mn><mrow><mn>7<\/mn><mi>\/<\/mi><mn>4<\/mn><\/mrow><\/msup><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:5.9776pt;padding-right:0pt;\"><mrow><mspace width=\"2em\"><\/mspace><mtext>\ud835\udc1e.)<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mrow><mroot><msup><mn>\ud835\udfd1<\/mn><mn>\ud835\udfd5<\/mn><\/msup><mn>\ud835\udfd2<\/mn><\/mroot><mo>=<\/mo><msup><mn>\ud835\udfd1<\/mn><mrow><mn>\ud835\udfd5<\/mn><mi>\/<\/mi><mn>\ud835\udfd2<\/mn><\/mrow><\/msup><\/mrow><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{array}{ll}\n\\sqrt{9} = 3 &amp; \\qquad \\textbf{If} \\quad \\mathbf{\\sqrt{9\\sqrt{9\\sqrt{9}}} = \\sqrt[4]{a}} \\\\[1em]\n\\sqrt{9\\sqrt{9}} = \\sqrt{9 \\times3} = \\sqrt{27} &amp; \\qquad a = \\left\\{ \\sqrt{9\\sqrt{9\\sqrt{9}}} \\right\\}^4 \\\\[1em]\n\\sqrt{9\\sqrt{9\\sqrt{9}}} = \\sqrt{9\\sqrt{27}} &amp; \\qquad \\left\\{ \\sqrt{9\\sqrt{9\\sqrt{9}}} \\right\\}^2 = 9\\sqrt{9\\sqrt{9}} \\\\[1em]\n&amp; \\qquad  \\left\\{ 9\\sqrt{9\\sqrt{9}} \\right\\}^2 = 9^2 \\times 9\\sqrt{9} = 81 \\times 9\\sqrt{9} = 729\\sqrt{9}  \\\\[1em]\n&amp; \\qquad  a = 729 \\times 3 = 2187  \\\\[2em]\n\\sqrt{9\\sqrt{9\\sqrt{9}}} = \\sqrt[4]{2187} \\mapsto 2187 = 3^7 \\\\[1em]\n\\sqrt[4]{2187} = \\sqrt[4]{3^7} = 3^{7\/4} &amp; \\qquad \\textbf{e.)} \\ \\ \\ \\ \\mathbf{\\sqrt[4]{3^7} = 3^{7\/4}}\n\\end{array}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div style=\"height:15px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div style=\"height:15px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Az \u2019a\u2019 \u00e9rt\u00e9ke valamilyen eg\u00e9sz sz\u00e1m legyen. Fontos tudni, hogy itt nem egy ismeretlent keres\u00fcnk, hanem egy meghat\u00e1rozott aritmetikai m\u0171veletsor ker\u00fcl kifejez\u00e9sre, \u00edgy az \u2019a\u2019 \u00e9rt\u00e9ke pozit\u00edv eg\u00e9sz sz\u00e1m lesz. Az\u00e9rt, mert pozit\u00edv eg\u00e9sz sz\u00e1mos gy\u00f6k ker\u00fcl kifejez\u00e9sre. Ne ess\u00fcnk abba a hib\u00e1ba, hogy n\u00e9gyzetre emel\u00e9sn\u00e9l \u2019\u00b1a\u2019-val sz\u00e1molunk.<\/p>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-1&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-1-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-1\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">English translation<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-1\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-1-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p class=\"wp-block-paragraph\">This problem was posted on Facebook by <a href=\"https:\/\/www.facebook.com\/matematica101\/reels\/\">>> digital content creator  Matem\u00e1tica Simples &lt;&lt;<\/a> in a <a href=\"https:\/\/www.facebook.com\/reel\/1684035009542254\">>> Reels video &lt;&lt;<\/a>.<br><br>I prepared the solution using the &#8222;lazy math&#8221; methodology. The magic, the simple solution to the problem lies in introducing a variable marked &#8216;a&#8217;. This helps to ensure that the solution requires very minimal use of a calculator. I took out my calculator about twice.<br><br>But I won&#8217;t drag it out any longer, let&#8217;s see the solution:<br><br>The value of &#8216;a&#8217; should be some integer. It is important to know that we are not looking for an unknown here, but a specific arithmetic sequence is being expressed, so the value of &#8216;a&#8217; will be a positive integer. This is because the root of a positive integer is being expressed. Let&#8217;s not make the mistake of calculating with &#8216;\u00b1a&#8217; when squaring.<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Ezt a feladatot, a Facebookon >> Matem\u00e1tica Simples digit\u00e1lis tartalomk\u00e9sz\u00edt\u0151 &lt;&lt; tette k\u00f6zz\u00e9 egy >> Reels vide\u00f3ban &lt;&lt;. A megold\u00e1st a &#8222;lusta matekos&#8221; metodik\u00e1j\u00e1val k\u00e9sz\u00edtettem el. A var\u00e1zslat, a feladat egyszer\u0171 megold\u00e1sa abban rejlik, hogy bevezet\u00fcnk egy &#8216;a&#8217;-val jelzett v\u00e1ltoz\u00f3t. Ez seg\u00edt abban, hogy a megold\u00e1shoz nagyon minim\u00e1lis sz\u00e1mol\u00f3g\u00e9p haszn\u00e1lat sz\u00fcks\u00e9ges. \u00c9n kb. k\u00e9tszer vettem [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":2181,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,5,2],"tags":[6,7,8],"class_list":["post-2174","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-gyokvonas","category-matematika","category-tudaster","tag-aritmetika","tag-gyokok","tag-gyokvonas"],"featured_image_urls":{"full":["https:\/\/okoster.hu\/tudaster\/wp-content\/uploads\/sites\/2\/2026\/06\/Emeletes-gyok.png",1376,768,false],"thumbnail":["https:\/\/okoster.hu\/tudaster\/wp-content\/uploads\/sites\/2\/2026\/06\/Emeletes-gyok-150x150.png",150,150,true],"medium":["https:\/\/okoster.hu\/tudaster\/wp-content\/uploads\/sites\/2\/2026\/06\/Emeletes-gyok-300x167.png",300,167,true],"medium_large":["https:\/\/okoster.hu\/tudaster\/wp-content\/uploads\/sites\/2\/2026\/06\/Emeletes-gyok-768x429.png",768,429,true],"large":["https:\/\/okoster.hu\/tudaster\/wp-content\/uploads\/sites\/2\/2026\/06\/Emeletes-gyok-1024x572.png",1024,572,true],"1536x1536":["https:\/\/okoster.hu\/tudaster\/wp-content\/uploads\/sites\/2\/2026\/06\/Emeletes-gyok.png",1376,768,false],"2048x2048":["https:\/\/okoster.hu\/tudaster\/wp-content\/uploads\/sites\/2\/2026\/06\/Emeletes-gyok.png",1376,768,false]},"author_info":{"info":["A-Ty"]},"category_info":"<a href=\"https:\/\/okoster.hu\/tudaster\/category\/gyokvonas\/\" rel=\"category tag\">gy\u00f6kvon\u00e1s<\/a> <a href=\"https:\/\/okoster.hu\/tudaster\/category\/matematika\/\" rel=\"category tag\">matematika<\/a> <a href=\"https:\/\/okoster.hu\/tudaster\/category\/tudaster\/\" rel=\"category tag\">Tud\u00e1st\u00e9r<\/a>","tag_info":"Tud\u00e1st\u00e9r","comment_count":"0","_links":{"self":[{"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/posts\/2174","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/comments?post=2174"}],"version-history":[{"count":3,"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/posts\/2174\/revisions"}],"predecessor-version":[{"id":2180,"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/posts\/2174\/revisions\/2180"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/media\/2181"}],"wp:attachment":[{"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/media?parent=2174"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/categories?post=2174"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/okoster.hu\/tudaster\/wp-json\/wp\/v2\/tags?post=2174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}