# Solve equations with exponents using the odd root property and the even root property

Another type of equation we can solve is one with exponents. As you might expect we can clear exponents by using roots. This is done with very few unexpected results when the exponent is odd. We solve these problems very straight forward using the odd root property

Odd Root Property: if an = b, then $a=\sqrt[n]{b}$ when n is odd

Example 1:

Solve x5 = 32

Use odd root property

$\sqrt[5]{x^5}=\sqrt[5]{32}$

x = 2 (our solution).

However, when the exponent is even we will have two results from taking an even root of both sides. One will be positive and one will be negative. This is because both 32 = 9 and ( – 3)2 = 9. so when solving x2 = 9 we will have two solutions, one positive and one negative: x = 3 and -3.

Even Root Property: if an = b, then $a=\pm \sqrt[n]{b}$ when n is even.

Example 2:

Solve x4 = 16

Use even root property ($\pm$)

$\sqrt[4]{x^4}=\sqrt[4]{16}$

$x=\pm 2$ (our solution).

Also Read : Solving quadratic Equations – which method?

World View Note: In 1545, French Mathematicain Gerolamo Cardano published his book The Great Art, or the Rules of Algebra which included the solution of an equation with a fourth power, but it was considered absurd by many to take a quantity to the fourth power because there are only three dimensions!.

Also Read : Solve equations with radicals and check for extraneous solution

Example 3:

Solve (2x + 4)2 = 36

Use even root property ( ± )

$\sqrt{(2x+4)^2}=\pm \sqrt{36}$

2x + 4 =± 6

To avoid sign errors we need two equations

2x + 4 =6 or 2x + 4 = -6

One equation for +, one equation for –

2x = 2 or 2x = -10

x = 1 or x = -5 (our solution).

In the previous example we needed two equations to simplify because when we took the root, our solutions were two rational numbers, 6 and – 6. If the roots did not simplify to rational numbers we can keep the ± in the equation.

Example 4:

Solve (6x – 9)2 = 45

Use even root property ( ±)

$\sqrt{(6x-9)^2}=\pm \sqrt{45}$

Simplify roots

$6x-9=\pm 3\sqrt{5}$

Use one equation because root did not simplify to rational

$6x=9\pm 3\sqrt{5}$

$x=\frac{9\pm 3\sqrt{5}}{6}$

Simplify, divide each termby 3

$x=\frac{3\pm \sqrt{5}}{2}$ (our solution).

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