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<html><table height="500" width="1000" border="2"><TR height="5" width="1000"><strong><center><marquee><font color="Green"><h1>APTITUDE</h1></center></strong></font></TR></marquee><TR><TD align="left" width="200" valign="top"><table><TR><a href="numbers.html"><strong>Numbers</strong></a></TR><br><TR><a href="hcf.html"><strong>H.C.F and L.C.M</strong></a></TR><br><TR><a href="dec.html" ><strong>Decimal Fractions</strong></a></TR><br><TR><a href="simplification.html"><strong>Simplification</strong></a></TR><br><TR><a href="squareandcuberoot.html" ><strong>Square and Cube roots</strong></a></TR><br><TR><a href="average.html" ><strong>Average</strong></a></TR><br><TR><a href="pnumbers.html" ><strong>Problems on Numbers</strong></a></TR><br><TR><a href="problemsonages.html"><strong>Problems on Ages</strong></a></TR><br><TR><a href="surdsandindices.html"><strong>Surds and Indices</strong></a></TR><br><TR><a href="percent.html" ><strong>Percentage</strong></a></TR><br><TR><a href="profitandloss.html" ><strong>Profit and Loss</strong></a></TR><br><TR><a href="ratioandproportion.html" ><strong>Ratio And Proportions</strong></a></TR><br><TR><a href="partnership.html"><strong>Partnership</strong></a></TR><br><TR><a href="chainrule1.html"><strong>Chain Rule</strong></a></TR><br><TR><a href="timeandwork.html" ><strong>Time and Work</strong></a></TR><br><TR><a href="pipesandcisterns.html" ><strong>Pipes and Cisterns</strong></a></TR><br><TR><a href="timeanddistance.html"><strong>Time and Distance</strong></a></TR><br><TR><a href="trains.html" ><strong>Trains</strong></a></TR><br><TR><a href="boats.html"><strong>Boats and Streams</strong></a></TR><br><TR><a href="alligation.html"><strong>Alligation or Mixture </strong></a></TR><br><TR><a href="simple.html" ><strong>Simple Interest</strong></a></TR><br><TR><a href="CI.html"><strong>Compound Interest</strong></a></TR><br><TR><a href=""><strong>Logorithms</strong></a></TR><br><TR><a href="areas.html" ><strong>Areas</strong></a></TR><br><TR><a href="volume.html"><strong>Volume and Surface area</strong></a></TR><br><TR><a href="races.html" ><strong>Races and Games of Skill</strong></a></TR><br><TR><a href="calendar.html" ><strong>Calendar</strong></a></TR><br><TR><a href="clocks.html" ><strong>Clocks</strong></a></TR><br><TR><a href="" ><strong>Stocks ans Shares</strong></a></TR><br><TR><a href="true.html" ><strong>True Discount</strong></a></TR><br><TR><a href="banker1.html" ><strong>Bankers Discount</strong></a></TR><br><TR><a href="oddseries.html"><strong>Oddmanout and Series</strong></a></TR><br><TR><a href=""><strong>Data Interpretation</strong></a></TR><br><TR><a href="probability.html"><strong>probability</strong></a></TR><br><TR><a href="percom1.html" ><strong>Permutations and Combinations</strong></a></TR><br><TR><a href="pinkivijji_puzzles.html" ><strong>Puzzles</strong></a></TR></table></TD><TD valign="top"><font size="5"><center><b>SQUAREROOTS AND CUBEROOTS</center></b></font><br><br><br><br><font size="4"> Formula:<br><br> The Product of two same numbers in easiest way as follow.<br><br>Example: let us calculate the product of 96*96<br><br>Solution: Here every number must be compare with the 100.<br> See here the given number 96 which is 4 difference with the 100.<br>so subtract 4 from the 96 we get 92 ,then the square of the number 4 <br>it is 16 place the 16 beside the 92 we get answer as 9216.<br><pre> 9 6<br> - 4<br> --------------<br> 9 2<br> --------------<br> 4*4=16<br> 9 2 1 6 <br></pre> therefore square of the two numbers 96*96=9216.<br><br><br>Example: Calculate product for 98*98<br><br>Solution: Here the number 98 is having 2 difference when compare to 100<br><br>subtract 2 from the number then we get 96 square the number 2 it is 4<br><br>now place beside the 96 as 9604<br><br><pre> 9 8 - 2 ------------- 9 6 ------------- 2*2=4 9 6 0 4.</pre> <br> so, we get the product of 98*98=9604.<br><br> Example : Calculate product for 88*88<br><br> Solution: Here the number 88 is having 12 difference when compare to 100<br>subtract 12 from the 88 then we get 76 <br>the square of the number 12 is 144 (which is three digit number but should<br> place only two digit beside the 76)<br>therefore in such case add one to 6 then it becomes 77 <br>now place 44 beside the number 77<br>we will get 7744 .<br>answer will be 7744.<br><pre> 88 -12 ------------ 76 ----------- 12*12=144 76 + 144 -------------------- 7744 -------------------- </pre><br><br>Example: Find the product of the numbers 46 *46?<br><br> Solution:<br> consider the number 50=100/2<br> now again go comparision with the number which gets<br> when division with 100.<br><br><br>here consider the number 50 which is nearer to the number given.<br><br>46 when compared with the number 50 we get the difference of 4.<br><br> now subtract the number 4 from the 46, we get 42.<br><br>as 50 got when 100 get divided by 2.<br><br>so, divided the number by 2 after subtraction.<br><br> 42/2=21<br><br>now square the the number 4 i.e, 4*4=16<br><br>just place the number 16 beside the number 21<br> we get 2116.<br><pre> 4 6 4 ---------------- 4 2 as 50 = 100/2 42/2=21</pre><br><br> now place 4*4=16 beside 21<br> 2 1 1 6<br> Answer will be 2116.<br><br><br>Example: Find the product of the numbers 37*37<br><br>Solution: <br> consider the number 50=100/2<br> now again go comparision with the number which gets when <br>division with 100.<br><br>here consider the number 50 which is nearer to the number given.<br><br><br> 37 when compared with the number 50 we get the difference of 13.<br> now subtract the number 13 from the 37, we get 24.<br> as 50 got when 100 get divided by 2.<br> so, divided the number by 2 after subtraction.<br> 24/2=12<br> now square the the number 13 i.e, 13*13=169<br> just place the number 169 beside the number 21<br><br> now as 169 is three digit number then add 1 to 2 we get <br><br>1t as 13 then place 69 beside the 13<br> we get 1369.<br><pre> 3 7 1 3 ----------------- 2 4 as 50 = 100/2 24/2=12 </pre><br> square 13* 13=169<br> <pre> 1 2 + 1 6 9 ----------------------- 1 3 6 9 -------------------------</pre><br><br><br>Example: Find the product of 106*106<br> <br> Solution: now compare it with 100 ,<br><br> The given number is more then 100<br> then add the extra number to the given number.<br> That is 106+6=112<br> then square the number 6 that is 6*6=36<br><br> just place beside the number 36 beside the 112,then <br> we get 11236.<br><br><pre> 1 0 6 + 6 --------------------- 1 1 2-------------------- </pre><br><br> now 6* 6=36 place this beside the number 112, we get<br> 1 1 2 3 6<br><br> Square root: If x2=y ,we say that the square root of y<br> is x and we write ,鈭歽=x.<br><br> Cube root : The cube root of a given number x is the number<br> whose cube is x. we denote the cube root of x by x1/3 .<br><br><br>Examples:<br> <br>1.Evaluate 6084<sup>1/2</sup> by factorization method.<br> <br> Solution: Express the given number as the product of prime <br>factors. Now, take the product of these prime factors choosing <br>one out of every pair of the same primes. This product gives the <br>square root of the given number.<br><br> Thus resolving 6084 in the prime factors ,we get 6084 <br> 2 6024<br> 2 3042<br> 3 1521<br> 3 507<br> 13 169<br> 13<br> <br><br> 6084=2<sup>1/2</sup> *3<sup>1/2</sup> *13<sup>1/2</sup><br><br><br> 6084<sup>1/2</sup>=2*3*13=78.<br><br> Answer is 78.<br><br><br><br>2.what will come in place of question mark in each of the following<br> questions?<br><br>i)(32.4/?)<sup>1/2</sup> = 2<br>ii)86.49<sup>1/2</sup> + (5+?<sup>1/2</sup>)<sup>2</sup> =12.3<br><br> Solution: 1) (32.4/x)<sup>1/2</sup>=2<br> Squaring on both sides we get<br> 32.4/x=4<br> =>4x=32.4<br> =>x=8.1<br><br> Answer is 8.1<br><br>ii)86.49<sup>1/2</sup> + 鈭⌨️ 快捷键说明
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