Trigonometry Examples

$3^{x^{2 - 12}} = 9^{2 x}$

Step 1

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

$3^{x^{2 - 12}} = 3^{2 (2 x)}$

Step 2

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

$x^{2 - 12} = 2 (2 x)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $x^{2 - 12}$ .

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Step 3.1.1

Rewrite.

$0 + 0 + x^{2 - 12} = 2 \cdot (2 x)$

Step 3.1.2

Simplify by adding zeros.

$x^{2 - 12} = 2 \cdot (2 x)$

Step 3.1.3

Subtract $12$ from $2$ .

$x^{- 10} = 2 \cdot (2 x)$

Step 3.1.4

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

$\frac{1}{x^{10}} = 2 \cdot (2 x)$

Step 3.2

Multiply $2$ by $2$ .

$\frac{1}{x^{10}} = 4 x$

Step 3.3

Subtract $4 x$ from both sides of the equation.

$\frac{1}{x^{10}} - 4 x = 0$

Step 3.4

Find the LCD of the terms in the equation.

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Step 3.4.1

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

$x^{10}, 1, 1$

Step 3.4.2

The LCM of one and any expression is the expression.

$x^{10}$

Step 3.5

Multiply each term in $\frac{1}{x^{10}} - 4 x = 0$ by $x^{10}$ to eliminate the fractions.

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Step 3.5.1

Multiply each term in $\frac{1}{x^{10}} - 4 x = 0$ by $x^{10}$ .

$\frac{1}{x^{10}} x^{10} - 4 x \cdot x^{10} = 0 x^{10}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

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

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Step 3.5.2.1.1.1

Cancel the common factor.

$\frac{1}{x^{10}} x^{10} - 4 x \cdot x^{10} = 0 x^{10}$

Step 3.5.2.1.1.2

Rewrite the expression.

$1 - 4 x \cdot x^{10} = 0 x^{10}$

Step 3.5.2.1.2

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

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Step 3.5.2.1.2.1

Move $x^{10}$ .

$1 - 4 (x^{10} x) = 0 x^{10}$

Step 3.5.2.1.2.2

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

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Step 3.5.2.1.2.2.1

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

$1 - 4 (x^{10} x^{1}) = 0 x^{10}$

Step 3.5.2.1.2.2.2

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

$1 - 4 x^{10 + 1} = 0 x^{10}$

Step 3.5.2.1.2.3

Add $10$ and $1$ .

$1 - 4 x^{11} = 0 x^{10}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

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

$1 - 4 x^{11} = 0$

Step 3.6

Solve the equation.

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Step 3.6.1

Subtract $1$ from both sides of the equation.

$- 4 x^{11} = - 1$

Step 3.6.2

Divide each term in $- 4 x^{11} = - 1$ by $- 4$ and simplify.

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Step 3.6.2.1

Divide each term in $- 4 x^{11} = - 1$ by $- 4$ .

$\frac{- 4 x^{11}}{- 4} = \frac{- 1}{- 4}$

Step 3.6.2.2

Simplify the left side.

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Step 3.6.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.6.2.2.1.1

Cancel the common factor.

$\frac{- 4 x^{11}}{- 4} = \frac{- 1}{- 4}$

Step 3.6.2.2.1.2

Divide $x^{11}$ by $1$ .

$x^{11} = \frac{- 1}{- 4}$

Step 3.6.2.3

Simplify the right side.

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Step 3.6.2.3.1

Dividing two negative values results in a positive value.

$x^{11} = \frac{1}{4}$

Step 3.6.3

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

$x = \sqrt[11]{\frac{1}{4}}$

Step 3.6.4

Simplify $\sqrt[11]{\frac{1}{4}}$ .

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Step 3.6.4.1

Rewrite $\sqrt[11]{\frac{1}{4}}$ as $\frac{\sqrt[11]{1}}{\sqrt[11]{4}}$ .

$x = \frac{\sqrt[11]{1}}{\sqrt[11]{4}}$

Step 3.6.4.2

Any root of $1$ is $1$ .

$x = \frac{1}{\sqrt[11]{4}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{\sqrt[11]{4}}$

Decimal Form:

$x = 0.88159125 \dots$