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Table A-11 Solution to Prob 7 in Chap 8 This shows that the multiplication-of-exponents rule applies to a power of a power of a power As you read down the left-hand column, each statement is equal to all the statements above it
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Statements (a p)q = a pq [(a p)q]r = (a pq)r [(a p)q]r = a (pq)r [(a p)q]r = a pqr QED Reasons GMOE rule as given in Chap 8 text, where a is any number except 0, and p and q are rational numbers Take rth power of both sides, where r is a rational number Consider (pq) as a single quantity and then use GMOE rule on right-hand side Ungrouping of products in exponent on right-hand side Mission accomplished
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7 This S/R proof is shown in Table A-11 We can legally take the r th power of both sides in line 2 of the proof even if r is a reciprocal power Remember, if there is any positivenegative ambiguity when taking a reciprocal power, the positive value is the default 8 We can start by stating the generalized multiplication-of-exponents (GMOE) rule as it appears in the chapter text The exponent names are changed to keep us out of a rote-memorization rut, and also to conform to the way the problem is stated For any number x except 0, and for any rational numbers r and s, (x r )s = x rs Applying the commutative law for multiplication to the entire exponent on the righthand side of this equation, we get (x r )s = x sr Finally, we can invoke the GMOE rule in reverse to the right-hand side, obtaining (x r )s = (x s )r QED Mission accomplished! 9 Let s suppose that x is a positive number It can by any number we want, as long as it is larger than 0 We take the 4th root or 1/4 power of x, and then square the result That gives us (x 1/4)2 According to the GMOE rule, that is the same as x (1/4) 2
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which simplifies to x 2/4 The fraction 2/4 can be reduced to 1/2 That means we actually have x raised to the 1/2 power, or the square root of x 10 Imagine that y is a positive number We take the 6th power of y, and then take the cube root or 1/3 power of the result That gives us (y 6)1/3 According to the GMOE rule, that is the same as y 6 (1/3) which simplifies to y 6/3 and then reduces to y 2
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1 We can suspect that quantity (d), 271/2, is irrational It is not a natural number When we use a calculator to evaluate it, we get 5196 followed by an apparently random jumble of digits That suggests its decimal expansion is endless and nonrepeating The other three quantities can be evaluated and found rational: (a) 163/4 = (161/4)3 = 23 = 8 (b) (1/4)1/2 = 1/(41/2) = 1/2 (c) ( 27) 1/3 = 1/( 271/3) = 1/( 3) = 1/3 2 If we have an irrational number expanded into endless, nonrepeating decimal form, we can multiply it by any natural-number power of 10 and always get the same string of digits The only difference will be that the decimal point moves to the right by one place for each power of 10 As an example, consider the square root of 7, or 71/2 Using a calculator with a large display, we see that this expands to 71/2 = 264575131106459059 As we multiply by increasing natural-number powers of 10, we get this sequence of numbers, each one 10 times as large as the one above it: 10 71/2 = 264575131106459059 100 71/2 = 264575131106459059 1,000 71/2 = 2,64575131106459059 10,000 71/2 = 26,4575131106459059 and so on, as long as we want These are all endless non-repeating decimals, so they re all irrational numbers This will happen for any endless non-repeating string of digits
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