Solving Equations with Radicals and Exponents
The EvenRoot Property
In solving the equation x^{2} = 4, you might be tempted to write x
= 2 as an equivalent
equation. But x = 2 is not equivalent to x^{2} = 4 because 2^{2}
= 4 and (2)^{2} = 4.
So the solution set to x^{2} = 4 is {2, 2}. The equation x^{2}
= 4 is equivalent to the
compound sentence x = 2 or x = 2, which we can abbreviate as x
= Â±2. The
equation x = Â±2 is read â€œx equals positive or negative 2.â€
Equations involving other even powers are handled like the squares. Because
2^{4} = 16 and (2)^{4} =16, the equation x^{4} = 16 is equivalent to x
= Â±2. So x^{4} = 16
has two real solutions. Note that x^{4} = 16 has no real solutions. The equation
x^{6} = 5 is equivalent to
We can now state a general rule.
EvenRoot Property
Suppose n is a positive even integer.
If k > 0, then x^{n} = k is equivalent to
.
If k = 0, then x^{n} = k is equivalent to x = 0.
If k
< 0, then x^{n} = k has no real solution.
Helpful hint
We do not say, â€œtake the
square root of each side.â€We
are not doing the same thing
to each side of x^{2} = 9 when
we write x = Â±3. Because there is only one odd
root of every real number, you
can actually take an odd root
of each side.
Example 1
Using the evenroot property
Solve each equation.
a) x^{2} = 10
b) w^{8} = 0
c) x^{4} = 4
Solution
a) x^{2} 
= 10 

x 

Evenroot property 
The solution set is {,
}, or
{Â±}.
b) w^{8} 
= 0 

w 
= 0 
Evenroot property 
The solution set is {0}.
c) By the evenroot property, x^{4} = 4 has no real solution. (The fourth power of
any real number is nonnegative.)
