Trigonometry Examples

$7^{x + 1} + 7^{x} + 7^{x - 1} = 399$

Step 1

Factor out $7^{x}$ from the expression.

$7^{x} (7 + 1 + \frac{1}{7})$

Step 2

Find the common denominator.

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Step 2.1

Write $7$ as a fraction with denominator $1$ .

$7^{x} (\frac{7}{1} + 1 + \frac{1}{7}) = 399$

Step 2.2

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

$7^{x} (\frac{7}{1} \cdot \frac{7}{7} + 1 + \frac{1}{7}) = 399$

Step 2.3

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

$7^{x} (\frac{7 \cdot 7}{7} + 1 + \frac{1}{7}) = 399$

Step 2.4

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

$7^{x} (\frac{7 \cdot 7}{7} + \frac{1}{1} + \frac{1}{7}) = 399$

Step 2.5

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

$7^{x} (\frac{7 \cdot 7}{7} + \frac{1}{1} \cdot \frac{7}{7} + \frac{1}{7}) = 399$

Step 2.6

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

$7^{x} (\frac{7 \cdot 7}{7} + \frac{7}{7} + \frac{1}{7}) = 399$

Step 3

Combine the numerators over the common denominator.

$7^{x} (\frac{7 \cdot 7 + 7 + 1}{7}) = 399$

Step 4

Simplify the expression.

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Step 4.1

Multiply $7$ by $7$ .

$7^{x} (\frac{49 + 7 + 1}{7}) = 399$

Step 4.2

Add $49$ and $7$ .

$7^{x} (\frac{56 + 1}{7}) = 399$

Step 4.3

Add $56$ and $1$ .

$7^{x} (\frac{57}{7}) = 399$

Step 5

Combine $7^{x}$ and $\frac{57}{7}$ .

$\frac{7^{x} \cdot 57}{7} = 399$

Step 6

Cancel the common factor of $7^{x}$ and $7$ .

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Step 6.1

Factor $7$ out of $7^{x} \cdot 57$ .

$\frac{7 (7^{x - 1} \cdot 57)}{7} = 399$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $7$ out of $7$ .

$\frac{7 (7^{x - 1} \cdot 57)}{7 (1)} = 399$

Step 6.2.2

Cancel the common factor.

$\frac{7 (7^{x - 1} \cdot 57)}{7 \cdot 1} = 399$

Step 6.2.3

Rewrite the expression.

$\frac{7^{x - 1} \cdot 57}{1} = 399$

Step 6.2.4

Divide $7^{x - 1} \cdot 57$ by $1$ .

$7^{x - 1} \cdot 57 = 399$

Step 7

Move $57$ to the left of $7^{x - 1}$ .

$57 \cdot 7^{x - 1} = 399$

Step 8

Divide each term in $57 \cdot 7^{x - 1} = 399$ by $57$ and simplify.

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Step 8.1

Divide each term in $57 \cdot 7^{x - 1} = 399$ by $57$ .

$\frac{57 \cdot 7^{x - 1}}{57} = \frac{399}{57}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $57$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{57 \cdot 7^{x - 1}}{57} = \frac{399}{57}$

Step 8.2.1.2

Divide $7^{x - 1}$ by $1$ .

$7^{x - 1} = \frac{399}{57}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $399$ by $57$ .

$7^{x - 1} = 7$

Step 9

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

$7^{x - 1} = 7^{1}$

Step 10

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

$x - 1 = 1$

Step 11

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

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Step 11.1

Add $1$ to both sides of the equation.

$x = 1 + 1$

Step 11.2

Add $1$ and $1$ .

$x = 2$