Algebra Know-It-ALL

4. The expression can be simplified to a sum of individual terms when we remember
   that squaring any quantity (that is, taking it to the second power) is the same thing as
   multiplying it by itself. Then we can use the results of the final “Challenge” in Chap. 5.
   Step-by-step, we get:

(y+ 1)^2 = ( y+ 1)( y+ 1)

=yy+y 1 + 1 y+ (1 × 1)

=y^2 +y+y+ 1

=y^2 + 2 y+ 1

5. This problem can be solved just like Prob. 4, but we have to pay careful attention
   because of the minus sign:

(y− 1)^2 = ( y− 1)( y− 1)

=yy+y(−1)+ (− 1 y)+ [(−1)× (−1)]

=y^2 + (−y)+ (−y)+ 1

=y^2 − 2 y+ 1

6. Let’s start with the generalized addition-of-exponents (GAOE) rule as it is stated in the
   chapter text. Here it is again, with the exponent names changed for variety! For any
   numbera except 0, and for any rational numbers p and q,

apaq=a(p+q)

Suppose r is another rational number. Let’s multiply both sides of the above equation by ar.

This gives us

(apaq)ar=a(p+q)ar

According to the rule for the grouping of factors in a product, we can take the parentheses

out of the left-hand side of the above equation and get

apaqar= a(p+ q)ar

The left-hand side of the equation now contains the expression we want to evaluate. Let’s

consider (p+q) to be a single quantity. We can then use the GAOE rule on the right-

hand side of this equation, getting

apaqar=a[(p+q)+r]

Again, we invoke the privilege of ungrouping, this time to the addends in the exponent

on the right-hand side. This gives us

apaqar=a(p+q+r)

Q.E.D. Mission accomplished!

Chapter 8 613