Precalculus Examples

$7^{\frac{x - 2}{6}} = \sqrt{7}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$7^{\frac{x - 2}{6}} = 7^{\frac{1}{2}}$

Step 2

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

$\frac{x - 2}{6} = \frac{1}{2}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $6$ .

$6 \frac{x - 2}{6} = 6 (\frac{1}{2})$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $6$ .

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Step 3.2.1.1.1

Cancel the common factor.

$6 \frac{x - 2}{6} = 6 (\frac{1}{2})$

Step 3.2.1.1.2

Rewrite the expression.

$x - 2 = 6 (\frac{1}{2})$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Factor $2$ out of $6$ .

$x - 2 = 2 (3) \frac{1}{2}$

Step 3.2.2.1.2

Cancel the common factor.

$x - 2 = 2 \cdot 3 \frac{1}{2}$

Step 3.2.2.1.3

Rewrite the expression.

$x - 2 = 3$

Step 3.3

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

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Step 3.3.1

Add $2$ to both sides of the equation.

$x = 3 + 2$

Step 3.3.2

Add $3$ and $2$ .

$x = 5$