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## Solve by Radical: xⁿ=c - Expii

#### Solving xn=c with prime factorization

Suppose c is a real number. We want to solve the equation and find the **real solutions** of xn=c

If n is odd, then there is always a unique real solution for x: x=n√c the nth root of c.

If n is even, we have to be more careful. First, there are no real solutions if c is negative. If c=0, there will just be the solution x=0. If 0">c>0, there will be a positive real solution and a negative real solution. x=±n√c

To simplify the nth root further when c is an integer (or sometimes a fraction), it helps to further break down c. Write c in terms of its prime factorization, we can then simplify.

For example, say c=240. We can use a prime factorization method to find: 240=2⋅2⋅2⋅2⋅3⋅5

Depending on what n is, x will be different.

If n=2, then:

x2=240√x2=±√240x=±√240x=±√2⋅2⋅2⋅2⋅3⋅5x=±4√15

So, the solutions of x2=240 are x=4√15 and x=−4√15. Note that both of these are real, not complex, solutions.

If n=3, then:

x3=2403√x3=3√240x=3√240x=3√2⋅2⋅2⋅2⋅3⋅5x=23√30

So, the real solution of x3=240 is x=23√30. (If you're curious, you can learn more about complex solutions on our site.)

Here is a graphic with the general method of solving by square roots and cube roots.

Image source: By Caroline Kulczycky

Try the following problem to test your understanding.

## Absolute value equations

### Absolute Value Equation entered :

|c-5|=7

### Step by step solution :

### Step 1 :

#### Rearrange this Absolute Value Equation

Absolute value equalitiy entered

|c-5| = 7

### Step 2 :

#### Clear the Absolute Value Bars

Clear the absolute-value bars by splitting the equation into its two cases, one for the Positive case and the other for the Negative case.

The Absolute Value term is |c-5|

For the Negative case we'll use -(c-5)

For the Positive case we'll use (c-5)

### Step 3 :

#### Solve the Negative Case

-(c-5) = 7

Multiply

-c+5 = 7

Rearrange and Add up

-c = 2

Multiply both sides by (-1)

c = -2

Which is the solution for the Negative Case

### Step 4 :

#### Solve the Positive Case

(c-5) = 7

Rearrange and Add up

c = 12

Which is the solution for the Positive Case

### Step 5 :

#### Wrap up the solution

c=-2

c=12

#### Solutions on the Number Line

### Two solutions were found :

- c=12
- c=-2

### Most Used Actions

\mathrm{simplify} | \mathrm{solve\:for} | \mathrm{expand} | \mathrm{factor} | \mathrm{rationalize} |

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