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Multiplying by 11
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BEDMAS & Fractions
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Writing Linear Equations in Slope-Intercept Form
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Simple Trinomials as Products of Binomials
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Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
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Coordinate System
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What the Factored Form of a Quadratic can tell you about the graph
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Solving Equations with Radicals and Exponents
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Writing Decimals as Fractions
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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 = ±2. The equation x = ±2 is read “x equals positive or negative 2.”

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 = ±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.


Helpful hint

We do not say, “take the square root of each side.”We are not doing the same thing to each side of x2 = 9 when we write x = ±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


a) x2 = 10  
x Even-root property

The solution set is {-, }, or {±}.

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.)

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