Trigonometry Examples

${(\frac{1}{16})}^{3 x - 4} = 64^{x - 1}$

Step 1

Apply the product rule to $\frac{1}{16}$ .

$\frac{1^{3 x - 4}}{16^{3 x - 4}} = 64^{x - 1}$

Step 2

One to any power is one.

$\frac{1}{16^{3 x - 4}} = 64^{x - 1}$

Step 3

Move $16^{3 x - 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$16^{- (3 x - 4)} = 64^{x - 1}$

Step 4

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

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

Step 5

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

$2 (- (3 x - 4)) = 3 (x - 1)$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 (- (3 x - 4))$ .

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Step 6.1.1

Rewrite.

$0 + 0 + 2 (- (3 x - 4)) = 3 (x - 1)$

Step 6.1.2

Simplify by adding zeros.

$2 (- (3 x - 4)) = 3 (x - 1)$

Step 6.1.3

Apply the distributive property.

$2 (- (3 x) - - 4) = 3 (x - 1)$

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $3$ by $- 1$ .

$2 (- 3 x - - 4) = 3 (x - 1)$

Step 6.1.4.2

Multiply $- 1$ by $- 4$ .

$2 (- 3 x + 4) = 3 (x - 1)$

Step 6.1.5

Apply the distributive property.

$2 (- 3 x) + 2 \cdot 4 = 3 (x - 1)$

Step 6.1.6

Multiply.

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Step 6.1.6.1

Multiply $- 3$ by $2$ .

$- 6 x + 2 \cdot 4 = 3 (x - 1)$

Step 6.1.6.2

Multiply $2$ by $4$ .

$- 6 x + 8 = 3 (x - 1)$

Step 6.2

Simplify $3 (x - 1)$ .

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Step 6.2.1

Apply the distributive property.

$- 6 x + 8 = 3 x + 3 \cdot - 1$

Step 6.2.2

Multiply $3$ by $- 1$ .

$- 6 x + 8 = 3 x - 3$

Step 6.3

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

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Step 6.3.1

Subtract $3 x$ from both sides of the equation.

$- 6 x + 8 - 3 x = - 3$

Step 6.3.2

Subtract $3 x$ from $- 6 x$ .

$- 9 x + 8 = - 3$

Step 6.4

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

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Step 6.4.1

Subtract $8$ from both sides of the equation.

$- 9 x = - 3 - 8$

Step 6.4.2

Subtract $8$ from $- 3$ .

$- 9 x = - 11$

Step 6.5

Divide each term in $- 9 x = - 11$ by $- 9$ and simplify.

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Step 6.5.1

Divide each term in $- 9 x = - 11$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{- 11}{- 9}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 9$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{- 11}{- 9}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{- 9}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Dividing two negative values results in a positive value.

$x = \frac{11}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{9}$

Decimal Form:

$x = 1.\overline{2}$

Mixed Number Form:

$x = 1 \frac{2}{9}$