Fractional Powers

3 Key excerpts on "Fractional Powers"

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  eBook - ePub

  Basic Math Concepts

  For Water and Wastewater Plant Operators

  • Joanne K. Price(Author)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  positive: 2 − 2 3 4 = 3 4 2 2 5 2 3 − 3 2 − 2 = 5 2 3 3 2 2 3 2 5 2 2 − 3 = 3 2 5 2 2 3 Example 2: (Negative Exponents) □ Write the following terms in expanded form using factors only (no. exponents): 10 x − 2 ; 2 5 • 4 − 2 ; 6 0 x 3 x − 2 10 x –2 = 10 x 2 = 10 (x) (x) 2 5 4 − 2 = 2 5 4 2 = (2) (2) (2) (2) (2) (4) (4) 6 0 x 3 x − 2 = 6 0 x 3 x 2 =[--=PLGO-SEPARATOR=--. ](1) (x) (x) (x) (x) (x) Example 3: (Negative Exponents) □ Complete the following calculations: (0.785)(60 2)(20); and. (3 2)(2 -3)(6 2) (0.785) (60 2) (20) = (0.785) (60) (60) (20) = 56, 520 3 2 2 − 3 6 2 = (3) (3) (6 3) (6 3) (2) (2) (2) = 40.5 ZERO EXPONENTS When a number has a zero exponent, it is always equal to one: 5 0 = 1 or x 0 = 1 DIVIDE. OUT FACTORS WHENEVER POSSIBLE As with other calculations, divide out common factors whenever possible. This makes the calculation less cumbersome. 13.3 FRACTIONAL EXPONENTS SUMMARY 1.  A root is a number which, when multiplied together two (or more) times, equals the original number. A square root is a number which, when multiplied together twice, equals the original number. For example, the square root of 64 is 8, since 8 × 8 = 64. A cube root is a number which, when multiplied together three times, equals the original number. For example, the cube root of 8 is 2, since 2 × 2 × 2 = 8. 2. A fractional exponent indicates a root is to be taken. The denominator of a fracitonal exponent determines which root is to be taken. 3. A fractional exponent of 1/2 indicates that a square root is to be taken. A square root may also be written as a radical: x 1 / 2 = x 4. A fractional exponent of 1/3 indicates that a cube root is to be taken. A cube root may also be written as a radical: x 1 / 3 = x 3 5.  The numerator of the fractional exponent is the power of the base. For example: This may be written using a radical as: 8 2 3 THE ROOT OF A NUMBER A root is a number which, when multiplied together a given number of times, equals the original number

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  eBook - ePub

  GED® Math Test Tutor, For the 2014 GED® Test

  • Sandra Rush(Author)
  • 2013(Publication Date)
  • Research & Education Association

    (Publisher)

  , so when dividing, subtract the exponents.

  3. Rule 3: (23 )2 = (8)2 = 64 = 26 , and 3 × 2 = 6, so when raising a power to a power, multiply the exponents.

  Roots

  Roots can be thought of as the opposite of powers. If 34 = 81, we say, “3 to the fourth power is 81.” The root sentence for this would be “The fourth root of 81 is 3,” and in symbols it is written as . So this “root” is actually answering the question, “What number repeated as a factor four times gives us 81?”

  The symbol for root is , called a radical, where the number for which we are finding a root (X in the radical below), called the radicand, goes under the radical sign and the root we are finding (n in the radical below), called the index, is inserted as a smaller (in size) number at the left of the radical sign.

  Because the most common index is 2, we don’t even bother to write the little 2 on the radical. We just know that has index 2, and it is called a square root. If the index is 3, we do write the little 3, so we have , which has the special name of cube root. All of the other roots are called by their indexes: “fourth root,” “fifth root,” and so forth.

  Fractional Exponents

  Fractional exponents imply roots. The numerator still indicates power, but the denominator of the fractional exponent indicates root. For example, means either (the cube root of 82 ) or , the square of the cube root of 8. It makes no difference whether you do the root or the power first. In this case, since the cube root of 8 is 2, it is easier to do the root first, and the answer is . Doing it the other way, we would have to remember , which, of course, also gives us 4.

  Roots of Negative Numbers

  You can find square roots of positive numbers only. The square root of a negative number is called imaginary because there are no two identical numbers that multiplied together will produce a negative number (two positives multiplied together or two negatives multiplied together will always produce a positive result). However, for odd roots, negative radicands are fine because three negatives multiplied together will produce a negative number—for example,

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  Mathematics for Biological Scientists

  • Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
  • 2009(Publication Date)
  • Garland Science

    (Publisher)

  nth root’ of a number.

  Finding the nth root of a number

  In much the same way as subtraction is the reverse of addition, and division the reverse of multiplication, taking a root is the reverse of raising to a power. For example, because we know that five to the power four is 625, we can say that a fourth root of 625 is five. We can write this using a special symbol as

  625 4

    = 5

  . Likewise we can write down that a square root of 25 is five:

  25 2

    = 5

  . Because we frequently need to write down the square root of a number, we usually omit the little ‘2’ before the square root symbol: for example 25 for the square root of 25. Note that there can be more than one square root or fourth root of a number. For instance if we square −5 we also get 25 and if we raise −5 to the fourth power we get 625, thus −5 is a square root of 25 and a fourth root of 625. We often write this by saying that the square roots of 25 are

  ± 25   =   ± 5

  and the real fourth roots of 625 are ±5. (If we allow complex numbers, there are two more fourth roots of 625. These are ±5i where

  i   =  

  − 1

  but we promised in the Introduction to ignore these.)

  Expressed generally if an = b, then a is an nth root of b. Whenever we need to specify only one nth root we require that the root be positive and call it the principal nth root,

  b n

  . Provided the original number, b, is positive there is always a principal nth root.

  Sometimes the principal root is reasonably obvious. For instance, 2 and −2 are square roots of 4 because either of these when squared gives 4. Similarly 3.1 is a square root of 9.61 (another is −3.1) because 3.12 = 9.61 and 3.1 is a cube root of 29.791 because 3.13 = 29.791. In each case the principal root is the positive value.

  However, sometimes the root is not obvious at all. Indeed sometimes it is impossible to write down an exact value of a principal root although for positive numbers we can always get as close as we like. Take a simple example: what is

  2 ?

  The principal square root of 1.9881, 1.9881 , is 1.41 and 2.0164 is 1.42 (you can check these by simple multiplication) but neither of these is 2 . We say that 2 is bracketed by the values 1.41 and 1.42. We can name the value, we just have, it is 2 , and we can always improve on the estimate by trial and error (see Box 2.1 ) but that only gives us a better approximation. In fact, we can prove that 2 is not a rational number (see Box 2.2 ). Any answer on a display or printout can only be an approximation. For instance a Casio FX-115W calculator gives

  2   = 1.414213562

  , but the square of this number is in fact closer to 1.999999999 than to 2, so the actual square root must be a little larger. The answer was the best the calculator could do. A more precise answer, 1.4142135623731, which can be coaxed out of Microsoft Excel®

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