Algebra Know-It-ALL

Part One 165

Answer 8-5

For any positive integer n, we call the nth root of a number the (1/n)th power of that number.
Then we insist that if n is even, the (1/n)th power is positive by default. We can specify that
we want to use the negative value instead of the positive one (or along with it, as we’ll do later
in this book when we solve quadratic equations), but we must be clear about it.

Question 8-6

What happens when we take a positive odd-integer root of a negative number? A positive
even-integer root of a negative number?

Answer 8-6

A positive odd-integer root of a negative number is another negative number. A positive even-
integer root of a negative number is an imaginary number. We haven’t worked with imaginary
numbers yet.

Question 8-7

Suppose we have a nonzero number x, and two integers p and q.We want to multiply xp by xq.
How can we express xpxq as a single power of x?

Answer 8-7

We add the exponents p and q. Then we raise x to that power, getting

xpxq=x(p+q)

Question 8-8

Suppose we have a nonzero number x, and two integers p and q. We want to divide xp by xq.
How can we express xp/xq as a single power of x?

Answer 8-8

We find the difference ( p−q). Then we raise x to that power, getting

xp/xq=x(p−q)

Question 8-9

Suppose we have a nonzero number x, and two integers p and q.We want to raise the quantity
xp to the qth power. How can we express (xp)q as a single power of x?

Answer 8-9

We multiply p by q. Then we raise x to that power, getting

(xp)q=xpq