![\"common](\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2019\/12\/mprep-blogimages-wave1-26-1-e1575498544522.png\")
<\/p>\n<\/p>\n
This post was written from the collective wisdom of Manhattan Prep GRE instructors.<\/span><\/i><\/p>\n Quick: here\u2019s an expression from a GRE math problem. How do you simplify it? <\/span><\/p>\n \u221a2<\/span>14<\/sup><\/span>+<\/span>2<\/span>17\u00a0<\/sup><\/span><\/p>\n A great GRE problem will often reveal math misconceptions we don\u2019t even know we have! There are a lot of different wrong ways to simplify this expression. Try it out before you keep reading\u2014then we\u2019ll look at some other math myths and common mistakes, and how to avoid second-guessing on the GRE.\u00a0<\/span><\/p>\n <\/p>\n First of all, if you added the exponents together, you\u2019re going down the wrong track. <\/span>Here are some things you can\u2019t do with exponents, and why.\u00a0<\/b><\/p>\n a\u00b2<\/span>\u00a0+ b\u00b2<\/span>\u00a0does NOT equal (a + b)\u00b2<\/span>\u00a0!<\/span><\/p>\n (a + b)\u00b2<\/span>\u00a0does NOT equal a\u00b2<\/span>\u00a0+ b\u00b2<\/span>!<\/span><\/p>\n a\u00b2<\/span>\u00a0<\/span>– b\u00b2<\/span>\u00a0does NOT equal (a – b)\u00b2<\/span>!<\/span><\/p>\n (a – b)\u00b2<\/span>\u00a0does NOT equal a\u00b2<\/span>\u00a0– b\u00b2<\/span>!<\/span><\/p>\n Double-check this rule using some small numbers. Remember the Pythagorean triples: sets of three integers that can be the side lengths of a right triangle, like 3, 4, and 5. Let <\/span>a<\/span><\/i> and <\/span>b<\/span><\/i> equal 3 and 4:\u00a0<\/span><\/p>\n 3\u00b2<\/span>\u00a0+ 4\u00b2<\/span>\u00a0= 5\u00b2<\/span>\u00a0= 25<\/span><\/p>\n (3 + 4)\u00b2<\/span>\u00a0= 7\u00b2<\/span>\u00a0= 49<\/span><\/p>\n So, you can\u2019t freely go back and forth between 3\u00b2<\/span>\u00a0+ 4\u00b2<\/span>, and (3 + 4)\u00b2<\/span>; they have different values.\u00a0<\/span><\/p>\n That\u2019s not exactly the situation in the problem from earlier, though. In that case, the two bases are the same, but the two exponents are different:\u00a0<\/span><\/p>\n 2<\/span>14<\/sup><\/span> + 2<\/span>17<\/sup><\/span><\/p>\n Unfortunately, <\/span>there\u2019s not a simple rule for adding exponents with the same base<\/b>, like there is for multiplying or dividing. The only way to actually simplify something like this is to <\/span>factor out a like term<\/b>.\u00a0<\/span><\/p>\n The number 2<\/span>14<\/sup><\/span> divides into both 2<\/span>14<\/sup><\/span> and 2<\/span>17<\/sup><\/span>. You can rewrite 2<\/span>14<\/sup><\/span> as (2<\/span>14<\/sup><\/span>)(1), and you can rewrite 2<\/span>17<\/sup><\/span> as (2<\/span>14<\/sup><\/span>)(2<\/span>3<\/span>). Here\u2019s how to start simplifying:\u00a0<\/span><\/p>\n \u221a2<\/span>14<\/sup><\/span>+<\/span>2<\/span>17<\/sup><\/span><\/p>\n \u221a(<\/span>2<\/span>14<\/sup><\/span>)(1) +<\/span>(2<\/span>14<\/sup><\/span>)<\/span>(2\u00b3<\/span>)<\/span><\/p>\n \u221a(<\/span>2<\/span>14<\/sup><\/span>)(1+<\/span>2\u00b3<\/span>)<\/span><\/p>\n \u221a(214<\/sup>)(9)<\/span><\/p>\n Finally, there\u2019s no more addition, so you can safely take the square root. The square root of 2<\/span>14<\/sup><\/span> is 2<\/span>7<\/sup><\/span>, and the square root of 9 is 3. So, the answer to the original problem is (2<\/span>7<\/sup><\/span>)(3).\u00a0<\/span><\/p>\n In short:\u00a0<\/span><\/p>\n By the way, the same goes for square roots. Here\u2019s something else you can\u2019t do with the expression above:\u00a0<\/span><\/p>\n \u221a2<\/span>14<\/sup><\/span>+<\/span>2<\/span>17<\/sup><\/span> does NOT equal \u221a<\/span>2<\/span>14<\/sup><\/span>+\u221a<\/span>2<\/span>17<\/sup><\/span>!<\/span><\/p>\n Use the same example, with 3, 4 and 5, to double-check. Because 3, 4, and 5 form a Pythagorean triple, \u221a<\/span>3\u00b2<\/span>+<\/span>4\u00b2<\/span>\u00a0=5<\/span>. That\u2019s not the same as \u221a<\/span>3\u00b2<\/span>+\u221a<\/span>4\u00b2<\/span>, which equals 7.\u00a0<\/span><\/p>\n When you see something unusual in an exponent\u2014a fraction, a negative number, or a variable\u2014the ultimate math mistake is to panic and bail out. <\/span>The normal exponent rules still work in the normal way<\/b>, even when the exponent looks strange. Apply the same rules that you would in an easier problem.\u00a0\u00a0<\/span><\/p>\n For example, the rules of exponents say that when you raise an exponent to another power, like (2<\/span>5<\/sup><\/span>)<\/span>7<\/sup><\/span>, you multiply the two exponents together, getting 2<\/span>35<\/sup><\/span>.\u00a0<\/span><\/p>\n Okay, what about (2<\/span>8x<\/sup><\/span>)<\/span>-0.5<\/sup><\/span>?\u00a0<\/span><\/p>\n Simplify it in exactly the same way as the previous problem: multiply the two exponents together, giving you 2<\/span>-4x<\/sup><\/span>.\u00a0<\/span><\/p>\n Normally, when you divide one exponent by another with the same base, you subtract the exponents from each other, like this:\u00a0<\/span><\/p>\n 2<\/span>10<\/sup>\/<\/span>2<\/span>4<\/sup><\/span>=<\/span>2<\/span>10-4<\/sup><\/span>=<\/span>2<\/span>6<\/sup><\/span><\/p>\n Use the same process when the exponents look awkward:\u00a0<\/span><\/p>\n 2<\/span>7x\/4<\/sup><\/span>\/<\/span>2<\/span>-x\/4<\/sup><\/span><\/p>\n 2<\/span>(7x\/4 – (-x\/4))<\/sup><\/span><\/p>\n1. Adding and Subtracting Exponents<\/b><\/h3>\n
\n
\n
\n
\n
2.\u00a0Weird Exponents<\/b><\/h3>\n