Trigonometry Examples

${(\frac{1}{4})}^{2 x + 2} = 64^{x - 1}$

Step 1

Apply the product rule to $\frac{1}{4}$ .

$\frac{1^{2 x + 2}}{4^{2 x + 2}} = 64^{x - 1}$

Step 2

One to any power is one.

$\frac{1}{4^{2 x + 2}} = 64^{x - 1}$

Step 3

Move $4^{2 x + 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4^{- (2 x + 2)} = 64^{x - 1}$

Step 4

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

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

Step 5

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

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

Step 6

Solve for $x$ .

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Step 6.1

Simplify $- (2 x + 2)$ .

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Step 6.1.1

Rewrite.

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

Step 6.1.2

Simplify by adding zeros.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $2$ by $- 1$ .

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

Step 6.1.4.2

Multiply $- 1$ by $2$ .

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

Step 6.2

Simplify $3 (x - 1)$ .

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Step 6.2.1

Apply the distributive property.

$- 2 x - 2 = 3 x + 3 \cdot - 1$

Step 6.2.2

Multiply $3$ by $- 1$ .

$- 2 x - 2 = 3 x - 3$

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.

$- 2 x - 2 - 3 x = - 3$

Step 6.3.2

Subtract $3 x$ from $- 2 x$ .

$- 5 x - 2 = - 3$

Step 6.4

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

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Step 6.4.1

Add $2$ to both sides of the equation.

$- 5 x = - 3 + 2$

Step 6.4.2

Add $- 3$ and $2$ .

$- 5 x = - 1$

Step 6.5

Divide each term in $- 5 x = - 1$ by $- 5$ and simplify.

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Step 6.5.1

Divide each term in $- 5 x = - 1$ by $- 5$ .

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

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 5$ .

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Step 6.5.2.1.1

Cancel the common factor.

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

Step 6.5.2.1.2

Divide $x$ by $1$ .

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

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.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.2$