Algebra Examples

$5^{x} + 5^{x - 1} = 150$

Step 1

Factor out $5^{x}$ from the expression.

$5^{x} (1 + \frac{1}{5})$

Step 2

Simplify the expression.

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Step 2.1

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

$5^{x} (\frac{5}{5} + \frac{1}{5}) = 150$

Step 2.2

Combine the numerators over the common denominator.

$5^{x} (\frac{5 + 1}{5}) = 150$

Step 2.3

Add $5$ and $1$ .

$5^{x} (\frac{6}{5}) = 150$

Step 3

Combine $5^{x}$ and $\frac{6}{5}$ .

$\frac{5^{x} \cdot 6}{5} = 150$

Step 4

Cancel the common factor of $5^{x}$ and $5$ .

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Step 4.1

Factor $5$ out of $5^{x} \cdot 6$ .

$\frac{5 (5^{x - 1} \cdot 6)}{5} = 150$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $5$ out of $5$ .

$\frac{5 (5^{x - 1} \cdot 6)}{5 (1)} = 150$

Step 4.2.2

Cancel the common factor.

$\frac{5 (5^{x - 1} \cdot 6)}{5 \cdot 1} = 150$

Step 4.2.3

Rewrite the expression.

$\frac{5^{x - 1} \cdot 6}{1} = 150$

Step 4.2.4

Divide $5^{x - 1} \cdot 6$ by $1$ .

$5^{x - 1} \cdot 6 = 150$

Step 5

Move $6$ to the left of $5^{x - 1}$ .

$6 \cdot 5^{x - 1} = 150$

Step 6

Divide each term in $6 \cdot 5^{x - 1} = 150$ by $6$ and simplify.

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Step 6.1

Divide each term in $6 \cdot 5^{x - 1} = 150$ by $6$ .

$\frac{6 \cdot 5^{x - 1}}{6} = \frac{150}{6}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $6$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{6 \cdot 5^{x - 1}}{6} = \frac{150}{6}$

Step 6.2.1.2

Divide $5^{x - 1}$ by $1$ .

$5^{x - 1} = \frac{150}{6}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $150$ by $6$ .

$5^{x - 1} = 25$

Step 7

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

$5^{x - 1} = 5^{2}$

Step 8

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

$x - 1 = 2$

Step 9

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

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Step 9.1

Add $1$ to both sides of the equation.

$x = 2 + 1$

Step 9.2

Add $2$ and $1$ .

$x = 3$