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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Equations with Radicals and Exponents

## The Even-Root Property

In solving the equation x2 = 4, you might be tempted to write x = 2 as an equivalent equation. But x = 2 is not equivalent to x2 = 4 because 22 = 4 and (-2)2 = 4. So the solution set to x2 = 4 is {-2, 2}. The equation x2 = 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 24 = 16 and (-2)4 =16, the equation x4 = 16 is equivalent to x = Â±2. So x4 = 16 has two real solutions. Note that x4 = -16 has no real solutions. The equation x6 = 5 is equivalent to We can now state a general rule.

Even-Root Property

Suppose n is a positive even integer.

If k > 0, then xn = k is equivalent to .

If k = 0, then xn = k is equivalent to x = 0.

If k < 0, then xn = k has no real solution.

We do not say, â€œtake the square root of each side.â€We are not doing the same thing to each side of x2 = 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 even-root property

Solve each equation.

a) x2 = 10

b) w8 = 0

c) x4 = -4

Solution

 a) x2 = 10 x Even-root property

The solution set is {-, }, or {Â±}.

 b) w8 = 0 w = 0 Even-root property

The solution set is {0}.

c) By the even-root property, x4 = -4 has no real solution. (The fourth power of any real number is nonnegative.)