Algebra Know-It-ALL

614 Worked-Out Solutions to Exercises: Chapters 1 to 9

7. This S/R proof is shown in Table A-11. We can “legally” take the rth power of both sides
   in line 2 of the proof even if r is a reciprocal power. Remember, if there is any positive-
   negative 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
   numberx except 0, and for any rational numbers r and s,

(xr)s=xrs

Applying the commutative law for multiplication to the entire exponent on the right-

hand side of this equation, we get

(xr)s=xsr

Finally, we can invoke the GMOE rule “in reverse” to the right-hand side, obtaining

(xr)s= (xs)r

Q.E.D. 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

(x1/4)^2

According to the GMOE rule, that is the same as

x(1/4)×^2

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.

Statements Reasons

(ap)q = apq GMOE rule as given in Chap. 8 text, where a is any

number except 0, and p and q are rational numbers

[(ap)q]r = (apq)r Take rth power of both sides, where r is a rational

number

[(ap)q]r = a(pq)r Consider (pq) as a single quantity and then use

GMOE rule on right-hand side

[(ap)q]r = apqr Ungrouping of products in exponent on right-hand

side

Q.E.D. Mission accomplished