212 algebra De mystif ieD

still struggling

Have you ever wondered why expressions such as^20 are not numbers? let us see

what complications arise when we try to see what^20 might mean. Say^20 =x.

2

01

=x

now cross-multiply.

2(1) = 0(x)

Multiplication by zero always yields zero, so the right-hand side is zero.

2 = 0 no value for x can make this equation true.

We could, instead, try to “clear the fraction” by multiplying both sides of the

equation by a common denominator. However, you will see that an absurd situ-

ation arises here, too.

02

0

⋅= 0 x

So, 0 = 0x, which is true for any x. actually, the expression^00 is not even defined

(what could it mean, anyway?).

?

On some equations, we need to raise both sides of the equation to a power

in order to solve for x. Be careful to raise both sides of the equation to the same

power, not simply the side with the root. Raising both sides of an equation to

an even power is another operation which can introduce extraneous solutions.

To see how this can happen, let’s look at the equation x = 4. If we square both

sides of the equation, we have the equation x^2 = 16. This equation has two

solutions: x = 4 and x = –4.

EXAMPLE

Solve the equation.

x− 1 = 6

Remember that ()a

2

= a if a is not negative. We will use this fact to pull

x – 1 from the square root sign. To “undo” a square root, first isolate the

EXAMPLE

Solve the equation.

EXAMPLE

Solve the equation.