Finite Math Examples

$\frac{1}{7} = \sqrt[7]{49^{x - 1}}$

Step 1

Rewrite the equation as $\sqrt[7]{49^{x - 1}} = \frac{1}{7}$ .

$\sqrt[7]{49^{x - 1}} = \frac{1}{7}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{49^{x - 1}}$ as $49^{\frac{x - 1}{7}}$ .

$49^{\frac{x - 1}{7}} = \frac{1}{7}$

Step 3

Raise $7$ to the power of $1$ .

$49^{\frac{x - 1}{7}} = \frac{1}{7^{1}}$

Step 4

Move $7^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$49^{\frac{x - 1}{7}} = 7^{- 1}$

Step 5

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

$7^{2 \frac{x - 1}{7}} = 7^{- 1}$

Step 6

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

$2 \frac{x - 1}{7} = - 1$

Step 7

Solve for $x$ .

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Step 7.1

Combine $2$ and $\frac{x - 1}{7}$ .

$\frac{2 (x - 1)}{7} = - 1$

Step 7.2

Multiply both sides of the equation by $\frac{7}{2}$ .

$\frac{7}{2} \cdot \frac{2 (x - 1)}{7} = \frac{7}{2} \cdot - 1$

Step 7.3

Simplify both sides of the equation.

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Step 7.3.1

Simplify the left side.

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Step 7.3.1.1

Simplify $\frac{7}{2} \cdot \frac{2 (x - 1)}{7}$ .

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Step 7.3.1.1.1

Cancel the common factor of $7$ .

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Step 7.3.1.1.1.1

Cancel the common factor.

$\frac{7}{2} \cdot \frac{2 (x - 1)}{7} = \frac{7}{2} \cdot - 1$

Step 7.3.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 (x - 1)) = \frac{7}{2} \cdot - 1$

Step 7.3.1.1.2

Cancel the common factor of $2$ .

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Step 7.3.1.1.2.1

Cancel the common factor.

$\frac{1}{2} (2 (x - 1)) = \frac{7}{2} \cdot - 1$

Step 7.3.1.1.2.2

Rewrite the expression.

$x - 1 = \frac{7}{2} \cdot - 1$

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

Simplify $\frac{7}{2} \cdot - 1$ .

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Step 7.3.2.1.1

Multiply $\frac{7}{2} \cdot - 1$ .

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Step 7.3.2.1.1.1

Combine $\frac{7}{2}$ and $- 1$ .

$x - 1 = \frac{7 \cdot - 1}{2}$

Step 7.3.2.1.1.2

Multiply $7$ by $- 1$ .

$x - 1 = \frac{- 7}{2}$

Step 7.3.2.1.2

Move the negative in front of the fraction.

$x - 1 = - \frac{7}{2}$

Step 7.4

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

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Step 7.4.1

Add $1$ to both sides of the equation.

$x = - \frac{7}{2} + 1$

Step 7.4.2

Write $1$ as a fraction with a common denominator.

$x = - \frac{7}{2} + \frac{2}{2}$

Step 7.4.3

Combine the numerators over the common denominator.

$x = \frac{- 7 + 2}{2}$

Step 7.4.4

Add $- 7$ and $2$ .

$x = \frac{- 5}{2}$

Step 7.4.5

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{2}$

Decimal Form:

$x = - 2.5$

Mixed Number Form:

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