Algebra Examples

$5^{3 x} \cdot 5 \cdot 5^{x + 1} = 5^{2 x + 3}$

Step 1

Multiply $5^{3 x}$ by $5$ .

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Step 1.1

Raise $5$ to the power of $1$ .

$5^{3 x} \cdot 5^{1} \cdot 5^{x + 1} = 5^{2 x + 3}$

Step 1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5^{3 x + 1} \cdot 5^{x + 1} = 5^{2 x + 3}$

Step 2

Multiply $5^{3 x + 1}$ by $5^{x + 1}$ by adding the exponents.

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Step 2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5^{3 x + 1 + x + 1} = 5^{2 x + 3}$

Step 2.2

Add $3 x$ and $x$ .

$5^{4 x + 1 + 1} = 5^{2 x + 3}$

Step 2.3

Add $1$ and $1$ .

$5^{4 x + 2} = 5^{2 x + 3}$

Step 3

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

$4 x + 2 = 2 x + 3$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $2 x$ from both sides of the equation.

$4 x + 2 - 2 x = 3$

Step 4.1.2

Subtract $2 x$ from $4 x$ .

$2 x + 2 = 3$

Step 4.2

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

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Step 4.2.1

Subtract $2$ from both sides of the equation.

$2 x = 3 - 2$

Step 4.2.2

Subtract $2$ from $3$ .

$2 x = 1$

Step 4.3

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Divide $x$ by $1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$