Precalculus Examples

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

Step 1

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

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

Step 2

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

$x = x^{2 - 12}$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Simplify $x^{2 - 12}$ .

Tap for more steps...

Step 3.1.1

Subtract $12$ from $2$ .

$x = x^{- 10}$

Step 3.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

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

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

Step 3.3

Find the LCD of the terms in the equation.

Tap for more steps...

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.

$1, x^{10}, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$x^{10}$

Step 3.4

Multiply each term in $x - \frac{1}{x^{10}} = 0$ by $x^{10}$ to eliminate the fractions.

Tap for more steps...

Step 3.4.1

Multiply each term in $x - \frac{1}{x^{10}} = 0$ by $x^{10}$ .

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

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Simplify each term.

Tap for more steps...

Step 3.4.2.1.1

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

Tap for more steps...

Step 3.4.2.1.1.1

Multiply $x$ by $x^{10}$ .

Tap for more steps...

Step 3.4.2.1.1.1.1

Raise $x$ to the power of $1$ .

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

Step 3.4.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4.2.1.1.2

Add $1$ and $10$ .

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

Step 3.4.2.1.2

Cancel the common factor of $x^{10}$ .

Tap for more steps...

Step 3.4.2.1.2.1

Move the leading negative in $- \frac{1}{x^{10}}$ into the numerator.

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

Step 3.4.2.1.2.2

Cancel the common factor.

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

Step 3.4.2.1.2.3

Rewrite the expression.

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

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Multiply $0$ by $x^{10}$ .

$x^{11} - 1 = 0$

Step 3.5

Solve the equation.

Tap for more steps...

Step 3.5.1

Add $1$ to both sides of the equation.

$x^{11} = 1$

Step 3.5.2

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

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

Step 3.5.3

Any root of $1$ is $1$ .

$x = 1$