Algebra Examples

$12^{2 x - 1} = {(\sqrt[4]{12})}^{x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{12}$ as $12^{\frac{1}{4}}$ .

$12^{2 x - 1} = {(12^{\frac{1}{4}})}^{x}$

Step 2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$12^{2 x - 1} = 12^{\frac{1}{4} x}$

Step 3

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

$2 x - 1 = \frac{1}{4} x$

Step 4

Solve for $x$ .

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Step 4.1

Simplify $\frac{1}{4} x$ .

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Step 4.1.1

Rewrite.

$2 x - 1 = 0 + 0 + \frac{1}{4} x$

Step 4.1.2

Simplify by adding zeros.

$2 x - 1 = \frac{1}{4} x$

Step 4.1.3

Combine $\frac{1}{4}$ and $x$ .

$2 x - 1 = \frac{x}{4}$

Step 4.2

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

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Step 4.2.1

Subtract $\frac{x}{4}$ from both sides of the equation.

$2 x - 1 - \frac{x}{4} = 0$

Step 4.2.2

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 x \cdot \frac{4}{4} - \frac{x}{4} - 1 = 0$

Step 4.2.3

Combine $2 x$ and $\frac{4}{4}$ .

$\frac{2 x \cdot 4}{4} - \frac{x}{4} - 1 = 0$

Step 4.2.4

Combine the numerators over the common denominator.

$\frac{2 x \cdot 4 - x}{4} - 1 = 0$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Simplify the numerator.

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Step 4.2.5.1.1

Factor $x$ out of $2 x \cdot 4 - x$ .

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Step 4.2.5.1.1.1

Factor $x$ out of $2 x \cdot 4$ .

$\frac{x (2 \cdot 4) - x}{4} - 1 = 0$

Step 4.2.5.1.1.2

Factor $x$ out of $- x$ .

$\frac{x (2 \cdot 4) + x \cdot - 1}{4} - 1 = 0$

Step 4.2.5.1.1.3

Factor $x$ out of $x (2 \cdot 4) + x \cdot - 1$ .

$\frac{x (2 \cdot 4 - 1)}{4} - 1 = 0$

Step 4.2.5.1.2

Multiply $2$ by $4$ .

$\frac{x (8 - 1)}{4} - 1 = 0$

Step 4.2.5.1.3

Subtract $1$ from $8$ .

$\frac{x \cdot 7}{4} - 1 = 0$

Step 4.2.5.2

Move $7$ to the left of $x$ .

$\frac{7 x}{4} - 1 = 0$

Step 4.3

Add $1$ to both sides of the equation.

$\frac{7 x}{4} = 1$

Step 4.4

Multiply both sides of the equation by $\frac{4}{7}$ .

$\frac{4}{7} \cdot \frac{7 x}{4} = \frac{4}{7} \cdot 1$

Step 4.5

Simplify both sides of the equation.

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Step 4.5.1

Simplify the left side.

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Step 4.5.1.1

Simplify $\frac{4}{7} \cdot \frac{7 x}{4}$ .

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Step 4.5.1.1.1

Cancel the common factor of $4$ .

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Step 4.5.1.1.1.1

Cancel the common factor.

$\frac{4}{7} \cdot \frac{7 x}{4} = \frac{4}{7} \cdot 1$

Step 4.5.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 x) = \frac{4}{7} \cdot 1$

Step 4.5.1.1.2

Cancel the common factor of $7$ .

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Step 4.5.1.1.2.1

Factor $7$ out of $7 x$ .

$\frac{1}{7} (7 (x)) = \frac{4}{7} \cdot 1$

Step 4.5.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 x) = \frac{4}{7} \cdot 1$

Step 4.5.1.1.2.3

Rewrite the expression.

$x = \frac{4}{7} \cdot 1$

Step 4.5.2

Simplify the right side.

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Step 4.5.2.1

Multiply $\frac{4}{7}$ by $1$ .

$x = \frac{4}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{7}$

Decimal Form:

$x = 0.\overline{571428}$