Precalculus Examples

e^{x} = e^{x^{2 - 12}}

$e^{x} = e^{x^{2 - 12}}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

e^{x} = e^{x^{2 - 12}}

$e^{x} = e^{x^{2 - 12}}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

x = x^{2 - 12}

$x = x^{2 - 12}$

Step 3

Solve for

x

$x$ .

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Step 3.1

Simplify

x^{2 - 12}

$x^{2 - 12}$ .

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Step 3.1.1

Subtract

12

$12$ from

2

$2$ .

x = x^{- 10}

$x = x^{- 10}$

Step 3.1.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

x = \frac{1}{x^{10}}

$x = \frac{1}{x^{10}}$

x = \frac{1}{x^{10}}

$x = \frac{1}{x^{10}}$

Step 3.2

Subtract

\frac{1}{x^{10}}

$\frac{1}{x^{10}}$ from both sides of the equation.

x - \frac{1}{x^{10}} = 0

$x - \frac{1}{x^{10}} = 0$

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

1, x^{10}, 1

$1, x^{10}, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

x^{10}

$x^{10}$

x^{10}

$x^{10}$

Step 3.4

Multiply each term in

x - \frac{1}{x^{10}} = 0

$x - \frac{1}{x^{10}} = 0$ by

x^{10}

$x^{10}$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in

x - \frac{1}{x^{10}} = 0

$x - \frac{1}{x^{10}} = 0$ by

x^{10}

$x^{10}$ .

x \cdot x^{10} - \frac{1}{x^{10}} x^{10} = 0 x^{10}

$x \cdot x^{10} - \frac{1}{x^{10}} x^{10} = 0 x^{10}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Multiply

x

$x$ by

x^{10}

$x^{10}$ by adding the exponents.

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Step 3.4.2.1.1.1

Multiply

x

$x$ by

x^{10}

$x^{10}$ .

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Step 3.4.2.1.1.1.1

Raise

x

$x$ to the power of

1

$1$ .

x^{1} x^{10} - \frac{1}{x^{10}} x^{10} = 0 x^{10}

$x^{1} x^{10} - \frac{1}{x^{10}} x^{10} = 0 x^{10}$

Step 3.4.2.1.1.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

x^{1 + 10} - \frac{1}{x^{10}} x^{10} = 0 x^{10}

$x^{1 + 10} - \frac{1}{x^{10}} x^{10} = 0 x^{10}$

x^{1 + 10} - \frac{1}{x^{10}} x^{10} = 0 x^{10}

$x^{1 + 10} - \frac{1}{x^{10}} x^{10} = 0 x^{10}$

Step 3.4.2.1.1.2

Add

1

$1$ and

10

$10$ .

x^{11} - \frac{1}{x^{10}} x^{10} = 0 x^{10}

$x^{11} - \frac{1}{x^{10}} x^{10} = 0 x^{10}$

x^{11} - \frac{1}{x^{10}} x^{10} = 0 x^{10}

$x^{11} - \frac{1}{x^{10}} x^{10} = 0 x^{10}$

Step 3.4.2.1.2

Cancel the common factor of

x^{10}

$x^{10}$ .

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Step 3.4.2.1.2.1

Move the leading negative in

- \frac{1}{x^{10}}

$- \frac{1}{x^{10}}$ into the numerator.

x^{11} + \frac{- 1}{x^{10}} x^{10} = 0 x^{10}

$x^{11} + \frac{- 1}{x^{10}} x^{10} = 0 x^{10}$

Step 3.4.2.1.2.2

Cancel the common factor.

x^{11} + \frac{- 1}{x^{10}} x^{10} = 0 x^{10}

$x^{11} + \frac{- 1}{x^{10}} x^{10} = 0 x^{10}$

Step 3.4.2.1.2.3

Rewrite the expression.

x^{11} - 1 = 0 x^{10}

$x^{11} - 1 = 0 x^{10}$

x^{11} - 1 = 0 x^{10}

$x^{11} - 1 = 0 x^{10}$

x^{11} - 1 = 0 x^{10}

$x^{11} - 1 = 0 x^{10}$

x^{11} - 1 = 0 x^{10}

$x^{11} - 1 = 0 x^{10}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Multiply

0

$0$ by

x^{10}

$x^{10}$ .

x^{11} - 1 = 0

$x^{11} - 1 = 0$

x^{11} - 1 = 0

$x^{11} - 1 = 0$

x^{11} - 1 = 0

$x^{11} - 1 = 0$

Step 3.5

Solve the equation.

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Step 3.5.1

Add

1

$1$ to both sides of the equation.

x^{11} = 1

$x^{11} = 1$

Step 3.5.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

x = \sqrt[11]{1}

$x = \sqrt[11]{1}$

Step 3.5.3

Any root of

1

$1$ is

1

$1$ .

x = 1

$x = 1$

x = 1

$x = 1$

x = 1

$x = 1$