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    How to get hold of roots and powers in GMAT Quant?

    Sayantani Barman Sayantani Barman
    Study Abroad Expert

    The GMAT exam conducted by GMAC has a global accreditation to it and that is for its universal acceptance. Throughout the world, the most reputed business schools accept GMAT scores. In fact, though in India CAT has been making the rounds for a long time now. But even a few top Indian business schools too, accept GMAT scores now. Now when we know how acclaimed this exam is, let us get into the GMAT syllabus which is very broad and quite obvious. GMAT has four sections – verbal reasoning which deals with the verbal and written English ability, logical reasoning, analytical writing assessment, and lastly quant section. Out of all these, the verbal ability and the quant section of GMAT are stated to be pretty straining. It is expected of the student to be proficient in these two genres considering the fact that they are interested in pursuing an MBA that too from a globally acclaimed institute. 

    When we are discussing the GMAT quant section, we must emphasize especially on the roots and powers heading which is dreaded by many students out there. No matter how well they prepare it, it seems that the topic is bewildering to many. Though it is generally stated that the GMAT quant section is synonymous to the high school syllabus but it is a quite difficult entrance exam where students have to showcase their knowledge. Let us today discuss a few of the basics of powers and roots which will enable the students to understand it better.

    Exponents

    Familiar with the term but completely forgotten about its content? Let us brush it up. Exponent is the number of times a particular number can be multiplied with itself. That number is known as the base number, for instance ‘k’, and the exponent ‘n’ states how many times that number can be multiplied with itself. Let us view how this whole thing will look –

    kn

    Now, looking at this figure we can recall the term ‘squares’ or k2 = k x k

    An exponent is not only limited to this but –

    22=2×2=4
    32=3×3=9
    42=4×4=16
    52=5×5=25

    23=2×2×2=8
    33=3×3×3=27
    43=4×4×4=64

    24=2×2×2×2=16
    34=3×3×3×3=81
    44=4×4×4×4=256
    54=5×5×5×5=625

    In the same way, exponents are also called ‘powers’ and are displayed by – kn as the nth power of k to the power of n. 

    The next term to be mentioned is squaring or when the number is raised to the power of 2. Raising the number to the power of 3 is called cubing.

    Roots

    The square root of any particular number, for instance, y is a number which when squared equals y. or in simpler terms, y = k2 and this means, k is the square root of y. In the same way, the cube root of any number says y, when cubed equals y or y = k3. 

    The square root of any number is symbolized by a symbol called ‘radical’ –

    4 = 2 

     25 = 5

    Any number which is beyond the square and cube roots are displayed in the superscript section like –

    38 = 2

    Features of exponents

    To understand this we need to further divide the features.

    Feature of exponent 1- 

    Any number having a power of 1 is just equivalent to itself. So, mostly no number is seen to have a power of 1. 

    21 = 2

    81 = 8

    Feature of exponent 0 –

    It is to be noted that any number having the power of 0 is equal to 1.

    20 = 1

    70 = 7

    When you square a number between 0 and 1

    As we have understood by now that when we are squaring any number 2 or more than that, the results will be higher than the original.

    When you are squaring to any number between 0 and 1, the base result will be a smaller number –

    (1/2)2 = ¼

    ¼ < ½

    Dealing with positive and negative powers

    Every positive number has two square roots – negative and positive respectively. This is because of a negative number, when multiplied with itself, results in a positive number. 

    4=2(−2)
    2×2=4
    (−2)×(−2)=4

    A cube or any other odd power has one particular root –

    This can be explained by the following –

    38 = 2
    23=8
    3-8=−2
    (−2)3=−8

    (-2)3 = -8

    It to be noted that old powers will always have the same sign as their roots.

    Negative squares

    While doing the square root of a negative number it has to be borne in mind that the square root if a negative number is not its real number.

    To explain this further,

    (-9) = not a real number

    And this is applicable to all even powers and so on.

    Negative exponents

    Any base number to a negative power is equivalent to 1 when divided by the base number to the positive type of the exponent –

    x-r = 1/xr 

    Here, r = positive integer and r = positive number.

    Explaining fractional exponents of GMAT Quant

    The name itself would make a student nervous but this explanation will help you solve questions like these every easily.

    Here, x is any positive number and r and s are any positive integers.

    Xr/s = (x1/s)r = (xr)1/s = 2r

    These two middle steps hereby explain the exponent to the power of another exponent. It is to be remembered that - xr/s = s√(xr)

    And let us now continue with the solution –

    27 2/3 = (271/3)2 = (272)1/3 = 3272 = 3729 = 9

    81/3 = 38 = 2

    91/2 = 9 = 3

    So, it can now be concluded that any number with a square root of ½ is the square root of that number and with cube root is the cube root of that number.

    Explaining the cycling property of GMAT Quant

    Consecutive powers consist of what is termed as cyclicity. If we take for instance the number 3 –

    31=3
    32=9
    33=27
    34=81
    35=243
    36=729

    If we look carefully we can notice that the last digit has repeated itself after a cycle of 4. So it is – 3, 9, 7, 1, and so on.

    This repetition when continues is called the cyclicity of numbers. This is very helpful when one needs to decode the unit digit of a given number. 

    Now when we have curated a few of the preparation tips for GMAT, it will be comparatively easier for the students to solve the same. 

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