Basic Math Examples

${(\frac{1}{19})}^{x - 1} = 19^{x}$

Step 1

Apply the product rule to $\frac{1}{19}$ .

$\frac{1^{x - 1}}{19^{x - 1}} = 19^{x}$

Step 2

One to any power is one.

$\frac{1}{19^{x - 1}} = 19^{x}$

Step 3

Move $19^{x - 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$19^{- (x - 1)} = 19^{x}$

Step 4

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

$- (x - 1) = x$

Step 5

Solve for $x$ .

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Step 5.1

Simplify $- (x - 1)$ .

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Step 5.1.1

Rewrite.

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

Step 5.1.2

Simplify by adding zeros.

$- (x - 1) = x$

Step 5.1.3

Apply the distributive property.

$- x - - 1 = x$

Step 5.1.4

Multiply $- 1$ by $- 1$ .

$- x + 1 = x$

Step 5.2

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

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Step 5.2.1

Subtract $x$ from both sides of the equation.

$- x + 1 - x = 0$

Step 5.2.2

Subtract $x$ from $- x$ .

$- 2 x + 1 = 0$

Step 5.3

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 5.4

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

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Step 5.4.1

Divide each term in $- 2 x = - 1$ by $- 2$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $- 2$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $x$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Dividing two negative values results in a positive value.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$