Trigonometry Examples

$5^{x} = {(\frac{1}{25})}^{4 x - 1}$

Step 1

Apply the product rule to $\frac{1}{25}$ .

$5^{x} = \frac{1^{4 x - 1}}{25^{4 x - 1}}$

Step 2

One to any power is one.

$5^{x} = \frac{1}{25^{4 x - 1}}$

Step 3

Move $25^{4 x - 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$5^{x} = 25^{- (4 x - 1)}$

Step 4

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

$5^{x} = 5^{2 (- (4 x - 1))}$

Step 5

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

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

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 (- (4 x - 1))$ .

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Multiply.

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Step 6.1.2.1

Multiply $4$ by $- 1$ .

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

Step 6.1.2.2

Multiply $- 1$ by $- 1$ .

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $- 4$ by $2$ .

$x = - 8 x + 2 \cdot 1$

Step 6.1.4.2

Multiply $2$ by $1$ .

$x = - 8 x + 2$

Step 6.2

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

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Step 6.2.1

Add $8 x$ to both sides of the equation.

$x + 8 x = 2$

Step 6.2.2

Add $x$ and $8 x$ .

$9 x = 2$

Step 6.3

Divide each term in $9 x = 2$ by $9$ and simplify.

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Step 6.3.1

Divide each term in $9 x = 2$ by $9$ .

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

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $9$ .

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Step 6.3.2.1.1

Cancel the common factor.

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

Step 6.3.2.1.2

Divide $x$ by $1$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{2}$