Trigonometry Examples

$81^{x + 2} = 27^{5 x + 4}$

Step 1

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

$3^{4 (x + 2)} = 3^{3 (5 x + 4)}$

Step 2

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

$4 (x + 2) = 3 (5 x + 4)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $4 (x + 2)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 4 (x + 2) = 3 (5 x + 4)$

Step 3.1.2

Simplify by adding zeros.

$4 (x + 2) = 3 (5 x + 4)$

Step 3.1.3

Apply the distributive property.

$4 x + 4 \cdot 2 = 3 (5 x + 4)$

Step 3.1.4

Multiply $4$ by $2$ .

$4 x + 8 = 3 (5 x + 4)$

Step 3.2

Simplify $3 (5 x + 4)$ .

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Step 3.2.1

Apply the distributive property.

$4 x + 8 = 3 (5 x) + 3 \cdot 4$

Step 3.2.2

Multiply.

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Step 3.2.2.1

Multiply $5$ by $3$ .

$4 x + 8 = 15 x + 3 \cdot 4$

Step 3.2.2.2

Multiply $3$ by $4$ .

$4 x + 8 = 15 x + 12$

Step 3.3

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.1

Subtract $15 x$ from both sides of the equation.

$4 x + 8 - 15 x = 12$

Step 3.3.2

Subtract $15 x$ from $4 x$ .

$- 11 x + 8 = 12$

Step 3.4

Move all terms not containing $x$ to the right side of the equation.

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Step 3.4.1

Subtract $8$ from both sides of the equation.

$- 11 x = 12 - 8$

Step 3.4.2

Subtract $8$ from $12$ .

$- 11 x = 4$

Step 3.5

Divide each term in $- 11 x = 4$ by $- 11$ and simplify.

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Step 3.5.1

Divide each term in $- 11 x = 4$ by $- 11$ .

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

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $- 11$ .

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Step 3.5.2.1.1

Cancel the common factor.

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

Step 3.5.2.1.2

Divide $x$ by $1$ .

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

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Move the negative in front of the fraction.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.\overline{36}$