{"id":2370,"date":"2024-06-19T12:00:58","date_gmt":"2024-06-19T19:00:58","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=2370"},"modified":"2020-01-15T10:50:35","modified_gmt":"2020-01-15T18:50:35","slug":"gmat-quant-roots","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-quant-roots\/","title":{"rendered":"GMAT Quant: Roots"},"content":{"rendered":"

This post was updated in 2024 for the new GMAT.<\/strong><\/p>\n

Master these seemingly intimidating mathematical symbols!<\/strong><\/p>\n

Practice Questions<\/h2>\n

First, try these practice questions.<\/p>\n

1) The numbers a, b, and c are all non-zero integers. Is a > 0?<\/p>\n

Statement #1: \"a=b^2\"
\nStatement #2: \u00a0\"a=sqrt{c}\"<\/p>\n

2) The numbers a, b, and c are all non-zero integers. Is a > 0?<\/p>\n

Statement #1: \"a=b^3\"
\nStatement #2: \u00a0\"a=root{3}{c}\"<\/p>\n

 <\/p>\n

Square Roots<\/h2>\n

When you square a number, you are multiplying it by itself, e.g. 6*6 = 36.\u00a0 When you take the square-root of a number, you are undoing the square, going backwards from the result of squaring to the input that was originally squared: \"sqrt{36}.\u00a0 Similarly, 8*8 = 64, so \"sqrt{64}.\u00a0 As long as all the numbers are positive, everything is straightforward.<\/p>\n

It’s easy to find the square root of a perfect square.\u00a0 All other square roots are ugly decimals.\u00a0 For estimation purposes on the very hardest GMAT questions, it might be useful to memorize that \"sqrt{2} and \"sqrt{3}, but without a calculator, no one is going to ask you to calculate the values of any decimal square roots bigger than that.\u00a0 If something like \"sqrt{52}\" shows up, all you have to recognize is between<\/em> what integers you would find that decimal.\u00a0 For example,<\/p>\n

\"49<\/p>\n

therefore<\/p>\n

\"sqrt{49}<\/p>\n

therefore<\/p>\n

\"7<\/p>\n

It’s also good to know how to simplify<\/a> square roots.<\/p>\n

<\/h2>\n

The Symbol: Positive or Negative?<\/h2>\n

\"\"<\/a><\/p>\n

 <\/p>\n

What is the name of this symbol?\u00a0 The benighted unwashed masses will call this simply the “square root symbol”, but that’s not the full story.\u00a0 The technical name is the “principal<\/span> square root symbol.” Here, “principal” (in the sense of “main” or “most important”) means: take the positive root only<\/strong>.<\/p>\n

This thickens the plot.\u00a0 The equation \"x^2 has two solutions, x = +4 and x = -4, because \"4^2 and \"(-4)^2, and the GMAT will impale you for only remembering one of those two.\u00a0 At the same time, \"sqrt{16}\" has only one output: \"sqrt{16} only.\u00a0 When you yourself undo a square by taking a square root, that’s a process that results in two possibilities, but when you see this symbol as such, printed as part of the problem, it means find the positive square root only.<\/p>\n

Notice that we can take the square root of zero: \"0^2=, so \"sqrt{0}, perfectly legal.\u00a0 Notice, also, we cannot<\/strong> take the square root of a negative — for example, \"sqrt{-1}\" — because that involves leaving the real number line.\u00a0 There are branches of math that do this, but it’s well beyond the scope of the GMAT.<\/p>\n

 <\/p>\n

Cubes and Cube Roots<\/h2>\n

When we raise a number to the third power, \"2^3\", that is called “cubing” it (because if we had a cube of side = 2, then the “cube” of that number would equal the volume of the cube).\u00a0 Here, \"2^3.\u00a0 A cube root simply undoes this process: \"root{3}{8}.\u00a0 As with a square roots, it’s easy to find the cube roots of perfect cubes, and on the GMAT you would never be expected to find an ugly decimal cube root without a calculator.<\/p>\n

Cubes and cube roots with negatives get interesting.\u00a0 While \"2^3, it turns out that \"(-2)^3.\u00a0\u00a0 When you multiply two negatives you get a positive, but when you multiply three negatives, you get a negative.\u00a0 More generally, when you multiply any even number of negatives you get a positive, but when you multiply any odd number of negatives, you get a negative.\u00a0 Therefore, when you cube a positive, you get a positive, but if you cube a negative, you get a negative.<\/p>\n

This means: while you can’t take the square root of a negative, you certainly can take the cube root of a negative.\u00a0 Undoing the equation \"(-2)^3, we get \"root{3}{-8}.\u00a0 In general, the cube root of a positive will be positive, and the cube root of a negative will be negative.<\/p>\n

It can also be a time-saver to remember the first five cubes:<\/p>\n

\"1^3<\/p>\n

\"2^3<\/p>\n

\"3^3<\/p>\n

\"4^3<\/p>\n

\"5^3<\/p>\n

You generally will not be expected to recognize cubes of larger numbers.\u00a0 Knowing just these will translate handily into all sorts of related facts: for example, \"root{3}{125} and \"root{3}{-125}.<\/p>\n

If you remember just this, you will be well-prepare for whatever the GMAT asks you about roots.<\/p>\n

 <\/p>\n

Additional Practice Question<\/h2>\n

3) http:\/\/gmat.magoosh.com\/questions\/91<\/a><\/p>\n

 <\/p>\n

Explanations of the Practice Questions<\/h2>\n

1) All that is given in the prompt is that a, b, and c are non-zero integers.<\/p>\n

Statement #1: the result of square anything is always positive, so whether b is negative or positive, a must be positive.\u00a0 This statement, by itself, is sufficient.<\/p>\n

Statement #2: since the square root symbol is printed as part of the problem, the output of the sqrt{c} must be positive.\u00a0 We know for a fact that a must be positive.\u00a0 Again, this statement, by itself, is sufficient.<\/p>\n

Both statements are sufficient.\u00a0 Answer = D<\/strong>.<\/p>\n

2) Again, all that is given in the prompt is that a, b, and c are non-zero integers.<\/p>\n

Statement #1: now, if we cube a positive, we get a positive, but if we cube a negative, we get a negative.\u00a0 The numbers a & b are either both positive or both negative, but since we don’t know the sign of b, we cannot determine the sign of a.\u00a0 This statement, by itself, is insufficient.<\/p>\n

Statement #2: if we take the cube root of a positive, we will get a positive, but if we take the cube-root of a negative, we get a negative.\u00a0 The numbers a & c are either both positive or both negative, but since we don’t know the sign of c, we cannot determine the sign of a.\u00a0 This statement, by itself, is insufficient.<\/p>\n

Combined Statements: If we put both statements together, we get that all three numbers, a, b, and c, have to have the same sign: either all three are positive, or all three are negative.\u00a0 We have no further information that would allow us to determine which of those two is the case.\u00a0 Thus, even with combined statements, we still do not have enough information to give a definitive answer to the prompt question.\u00a0 Combined, the statements are still insufficient.\u00a0 Answer = E<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

This post was updated in 2024 for the new GMAT. Master these seemingly intimidating mathematical symbols! Practice Questions First, try these practice questions. 1) The numbers a, b, and c are all non-zero integers. Is a > 0? Statement #1: Statement #2: \u00a0 2) The numbers a, b, and c are all non-zero integers. Is […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[150],"tags":[],"ppma_author":[13209],"class_list":["post-2370","post","type-post","status-publish","format-standard","hentry","category-basics"],"acf":[],"yoast_head":"\nGMAT Quant: Roots - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/gmat-quant-roots\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GMAT Quant: Roots\" \/>\n<meta property=\"og:description\" content=\"This post was updated in 2024 for the new GMAT. Master these seemingly intimidating mathematical symbols! 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Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":13209,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2370","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=2370"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/2370\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=2370"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=2370"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=2370"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=2370"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}