Trigonometry Examples

$16^{3 x} = 64^{- 3 x - 1}$

Step 1

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

$4^{2 (3 x)} = 4^{3 (- 3 x - 1)}$

Step 2

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

$2 (3 x) = 3 (- 3 x - 1)$

Step 3

Solve for $x$ .

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Step 3.1

Multiply $3$ by $2$ .

$6 x = 3 (- 3 x - 1)$

Step 3.2

Simplify $3 (- 3 x - 1)$ .

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Step 3.2.1

Apply the distributive property.

$6 x = 3 (- 3 x) + 3 \cdot - 1$

Step 3.2.2

Multiply.

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Step 3.2.2.1

Multiply $- 3$ by $3$ .

$6 x = - 9 x + 3 \cdot - 1$

Step 3.2.2.2

Multiply $3$ by $- 1$ .

$6 x = - 9 x - 3$

Step 3.3

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

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Step 3.3.1

Add $9 x$ to both sides of the equation.

$6 x + 9 x = - 3$

Step 3.3.2

Add $6 x$ and $9 x$ .

$15 x = - 3$

Step 3.4

Divide each term in $15 x = - 3$ by $15$ and simplify.

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Step 3.4.1

Divide each term in $15 x = - 3$ by $15$ .

$\frac{15 x}{15} = \frac{- 3}{15}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $15$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{15 x}{15} = \frac{- 3}{15}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{15}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Cancel the common factor of $- 3$ and $15$ .

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Step 3.4.3.1.1

Factor $3$ out of $- 3$ .

$x = \frac{3 (- 1)}{15}$

Step 3.4.3.1.2

Cancel the common factors.

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Step 3.4.3.1.2.1

Factor $3$ out of $15$ .

$x = \frac{3 \cdot - 1}{3 \cdot 5}$

Step 3.4.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot - 1}{3 \cdot 5}$

Step 3.4.3.1.2.3

Rewrite the expression.

$x = \frac{- 1}{5}$

Step 3.4.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{5}$

Decimal Form:

$x = - 0.2$