contemp. proj.

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FRACTIONAL EXPONENTS Keiko C alais Jenica alexis B ersosa


OBJECTIVES Fractional exponents is also called as radicals. This section aims to: -define radicals and illustrate its properties; -relate powers with rational exponents to radical form; and -simplify radicals by removing of perfect powers, reducing the given index, and rationalizing the denominator of the radicand. In this section, we shall extend the definition of a n to include all fractions or rational numbers for n.

Fractional exponents:

Fractional exponents Fractional exponents obey the same laws as do integral exponents. For example, Another way of expressing this would be Observe that the number 41/2 , when squared in the foregoing example, produced the number 4 as an answer. Recalling that a square root of a number N is a number x such that x2 = N, we conclude that 41/2 is equivalent to a. Thus we have a definition, as follows: A fractional exponent of the form 1/r indicates a root, the index of which is r. This is further illustrated in the following examples:

Fractional exponents:

Fractional exponents Notice that in an expression such as 8 2/3 we can either find the cube root of 8 first or square 8 first, as shown by the following example : All the numbers in the evaluation of 82/3 remain small if the cube root is found before raising the number to the second power. This order of operation is particularly desirable in evaluating a number like 645/6, If 64 were first raised to the fifth power, a large number would result. It would require a great deal of unnecessary effort to find the sixth root of 645. The result is obtained easily, if we write If an improper fraction occurs in an exponent, such as 7/3 in the expression 27/3 , it is customary to keep the fraction in that form rather than express it as a mixed number. In fraction form an exponent shows immediately what power is intended and what root is intended. However, 27/3 can be expressed in another form and simplified by changing the improper fraction to a mixed number and writing the fractional part in the radical form as follows:

Fractional exponents(lesson proper):

Fractional exponents(lesson proper ) EXPONENTS The exponent of a number says how many times to use the number in a multiplication. In this example: 82 = 8 × 8 = 64 In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared“ But what if the exponent is a fraction ? In the example above, the exponent was "2", but what if it were "½" ? How does that work? Question : What is x½ ? Answer : x ½ = the square root of x ( i.e. x½ = √ x) WHY? Because if you square x½ you get: (x½)2 = x1 = x Example: √2 × √2 = 2 Is also: 2 ½ × 2½ = 21

Fractional exponents(lesson proper):

Fractional exponents(lesson proper) To understand that, follow this two-step argument : 1. First , there is the general rule: ( xm )n = xm×n Example: (x2)3 = (xx)3 = (xx)(xx)(xx) = xxxxxx = x6 So (x2)3 = x2×3 = x6 2. Now , let's look at what happens when we square x½: ( x½)2 = x½×2 = x1 = x When we square x½ we get x, so x½ must be the square root of x Try Another Fraction Let us try that again, but with an exponent of one-quarter (1/4): What is x ¼ ? (x ¼ ) 4 = x ¼×4 = x 1 = x So, what value can be multiplied 4 times to get x? Answer: The fourth root of x. So, x ¼ = The 4th Root of x General Rule It worked for ½, it worked with ¼, in fact it works generally: x1/n = The n- th Root of x


FRACTIONAL EXPONENTS(LESSON PROPER) What About More Complicated Fractions? What about a fractional exponent like 4 3/2 ? That is really saying to do a cube (3) and a square root (1/2), in any order. Let me explain. A fraction (like m/n ) can be broken into two parts: a whole number part ( m ) , and a fraction ( 1/n ) part So, because m/n = m × (1/n) we can do this: The order does not matter, so it also works for m/n = (1/n) × m :


SOME EXAMPLES Some examples: Example: What is 4 3/2 ? 4 3/2 = 4 3×(1/2) = √(4 3 ) = √(4×4×4) = √(64) = 8 or 4 3/2 = 4 (1/2)×3 = (√4) 3 = (2) 3 = 8 Either way gets the same result. Example: What is 27 4/3 ? 27 4/3 = 27 4×(1/3) = (27 4 ) = (531441) = 81 or 27 4/3 = 27 (1/3)×4 = (27) 4 = (3) 4 = 81 It was certainly easier the 2nd way!

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