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exponents p and q be any rational numbers we want That gives us the powerful, far-reaching generalized multiplication-of-exponents (GMOE) rule ! If a, p, and q are rational numbers and a 0, then (a p)q = a pq We now have a way to evaluate an expression where we raise a number to a certain power, and then take a root of the result Remember that a root is a reciprocal power So, if we encounter an exponent that takes the form r /s, we can call this the product of r and 1/s, and then use the GMOE rule: (a r )1/s = a r (1/s) = a r /s That s how we d evaluate the sth root of a r But it also tells us something more: when we take a base number to a rational-number, noninteger power, it s the same thing as taking the base to an integer power and then taking an integer root of the result Remember, a rational number is a quotient of two integers! If we reverse the order of the terms in the above three-way equation, we get a r /s = a r (1/s) = (a r )1/s This is a heavy dose of abstract math! Let s look at a couple of specific cases where integers are raised to rational-number powers First, this: 106/3 = 106 (1/3) = (106)1/3 = 1,000,0001/3 = 100 If you re astute, you can solve this a lot quicker by noting that 6/3 = 2, so 106/3 = 102 = 100 Usually, rational-number powers aren t this easy to evaluate The results often produce numbers that aren t even rational Consider this example: 23/2 = 23 (1/2) = (23)1/2 = 81/2 = 28284 This is an endless nonrepeating decimal It cannot be expressed as a ratio of integers, and is not a rational number You ll learn more about these types of numbers in the next chapter
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Let s review the most important points in this chapter They can be condensed into six statements If a is any nonzero number and n is a negative integer, the expression a n means you should raise a to the power of |n|, and then take the reciprocal of the result If a is any nonzero number and m and n are rational numbers, then a m times a n is the same as a raised to the power of (m + n) If a is any nonzero number and m and n are rational numbers, then a m divided by a n is the same as a raised to the power of (m n) If a is any nonzero number and p is any nonzero integer, then the pth root of a is the same as raising a to the power of 1/p If a, p, and q are rational numbers and a is nonzero, then if you raise a to the pth power and take the result to the qth power, it s the same as raising a to the power of pq If a, p, and q are rational numbers with a and q nonzero, then if you raise a to the pth power and take
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the qth root of the result, it s the same as raising a to the power of p /q
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What do you get if you take the 5/2 power of 6 Mathematically, evaluate this expression and use a calculator to figure out the result to several decimal places: 6( 5/2)
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Let s apply the GMOE rule to this problem It can be tricky because of the minus sign, and we have to be sure we remember the difference between negative powers and reciprocal powers Let s go: 6( 5/2) = 6 5 (1/2) = (6 5)1/2 = [1/(65)]1/2 = (1/7,776)1/2 Now it s time to use a calculator! Remember that the 1/2 power is the same as the square root First we take the reciprocal of 7,776, getting a decimal point, three ciphers, and a long string of digits Then we hit the square root key with the string of digits still in the display, getting 001134023 The digits go on without end, and there s no apparent pattern As things turn out, this is not a rational number
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