Algebra Know-It-ALL

Quotient of powers

For all nonzero real numbers x, y, andz,

xy /z= (xy)^1 /z= (x1/z)y

Power of product

For all nonzero real numbers x, y, andz,

(xy)z=xzyz

Power of quotient

For all nonzero real numbers x, y, and z,

(x /y)z=xz/yz

Power of reciprocal

Letx be a nonzero real number. Let y be any real number. Then

(1/x)y= 1/(xy)

Square of sum

For all real numbers x and y,

(x+y)^2 =x^2 + 2 xy+y^2

Square of difference

For all real numbers x and y,

(x−y)^2 =x^2 − 2 xy+y^2

Are you confused?

Some of the rules above involve real-number exponents. That can include irrationals! You may wonder
how can anybody define such a thing as a number raised to the power of pi (π), for example. You’ll under-
stand irrational exponents better when you learn about logarithmic and exponential functions in Part 3.
For now, here’s a little glimpse into the mystery.
Imagine that you stumble across the expression 2π in some mathematical adventure. The value of π, to
five decimal places, is 3.14159, but you know it can’t be expressed exactly as a ratio of integers, because it

More Rules for Real Variables 139