{"id":22540,"date":"2025-06-16T19:40:55","date_gmt":"2025-06-16T19:40:55","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=22540"},"modified":"2025-06-16T19:40:57","modified_gmt":"2025-06-16T19:40:57","slug":"the-square-root-of-209-lies-between","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-square-root-of-209-lies-between\/","title":{"rendered":"The square root of 209 lies between"},"content":{"rendered":"\n<p>The square root of 209 lies between <\/p>\n\n\n\n<p>13 and 14 <\/p>\n\n\n\n<p>14 and 15 <\/p>\n\n\n\n<p>15 and 16 <\/p>\n\n\n\n<p>16 and 17<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p>The square root of 209 lies <strong>between 14 and 15<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>To find between which two whole numbers the square root of 209 lies, we compare 209 to perfect squares of whole numbers:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>132=16913^2 = 169<\/li>\n\n\n\n<li>142=19614^2 = 196<\/li>\n\n\n\n<li>152=22515^2 = 225<\/li>\n\n\n\n<li>162=25616^2 = 256<\/li>\n\n\n\n<li>172=28917^2 = 289<\/li>\n<\/ul>\n\n\n\n<p>We see that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>142=19614^2 = 196 which is <strong>less than 209<\/strong><\/li>\n\n\n\n<li>152=22515^2 = 225 which is <strong>greater than 209<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Therefore, the square root of 209 must be <strong>greater than 14 but less than 15<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">More detailed reasoning:<\/h3>\n\n\n\n<p>The square root function is increasing and continuous. For any number xx, if n2&lt;x&lt;(n+1)2n^2 &lt; x &lt; (n+1)^2, then x\\sqrt{x} lies between nn and n+1n+1.<\/p>\n\n\n\n<p>Here, since 196&lt;209&lt;225196 &lt; 209 &lt; 225, it follows that: 14&lt;209&lt;1514 &lt; \\sqrt{209} &lt; 15<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Estimating the value:<\/h3>\n\n\n\n<p>To approximate 209\\sqrt{209}, consider how far 209 is from 196 and 225:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Distance from 196 to 209 is 209\u2212196=13209 &#8211; 196 = 13<\/li>\n\n\n\n<li>Distance from 196 to 225 is 225\u2212196=29225 &#8211; 196 = 29<\/li>\n<\/ul>\n\n\n\n<p>The difference between the squares is 29, and 209 is 13 units above 196, so the square root of 209 is roughly: 14+1329\u224814+0.45=14.4514 + \\frac{13}{29} \\approx 14 + 0.45 = 14.45<\/p>\n\n\n\n<p>Thus, 209\u224814.45\\sqrt{209} \\approx 14.45.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Summary:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Since 142=196&lt;209&lt;225=15214^2 = 196 &lt; 209 &lt; 225 = 15^2, the square root of 209 lies between 14 and 15.<\/li>\n\n\n\n<li>A rough estimate places it near 14.45.<\/li>\n\n\n\n<li>This method works for any number by comparing to perfect squares and finding approximate location between consecutive integers.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The square root of 209 lies between 13 and 14 14 and 15 15 and 16 16 and 17 The correct answer and explanation is: The square root of 209 lies between 14 and 15. Explanation: To find between which two whole numbers the square root of 209 lies, we compare 209 to perfect squares [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-22540","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22540","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=22540"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22540\/revisions"}],"predecessor-version":[{"id":22541,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22540\/revisions\/22541"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=22540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=22540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=22540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}