Algebra Examples

$4^{x} - 4^{x - 1} = 24$

Step 1

Factor out $4^{x}$ from the expression.

$4^{x} (1 - \frac{1}{4})$

Step 2

Simplify the expression.

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Step 2.1

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

$4^{x} (\frac{4}{4} - \frac{1}{4}) = 24$

Step 2.2

Combine the numerators over the common denominator.

$4^{x} (\frac{4 - 1}{4}) = 24$

Step 2.3

Subtract $1$ from $4$ .

$4^{x} (\frac{3}{4}) = 24$

Step 3

Combine $4^{x}$ and $\frac{3}{4}$ .

$\frac{4^{x} \cdot 3}{4} = 24$

Step 4

Cancel the common factor of $4^{x}$ and $4$ .

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Step 4.1

Factor $4$ out of $4^{x} \cdot 3$ .

$\frac{4 (4^{x - 1} \cdot 3)}{4} = 24$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $4$ out of $4$ .

$\frac{4 (4^{x - 1} \cdot 3)}{4 (1)} = 24$

Step 4.2.2

Cancel the common factor.

$\frac{4 (4^{x - 1} \cdot 3)}{4 \cdot 1} = 24$

Step 4.2.3

Rewrite the expression.

$\frac{4^{x - 1} \cdot 3}{1} = 24$

Step 4.2.4

Divide $4^{x - 1} \cdot 3$ by $1$ .

$4^{x - 1} \cdot 3 = 24$

Step 5

Move $3$ to the left of $4^{x - 1}$ .

$3 \cdot 4^{x - 1} = 24$

Step 6

Divide each term in $3 \cdot 4^{x - 1} = 24$ by $3$ and simplify.

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Step 6.1

Divide each term in $3 \cdot 4^{x - 1} = 24$ by $3$ .

$\frac{3 \cdot 4^{x - 1}}{3} = \frac{24}{3}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{3 \cdot 4^{x - 1}}{3} = \frac{24}{3}$

Step 6.2.1.2

Divide $4^{x - 1}$ by $1$ .

$4^{x - 1} = \frac{24}{3}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $24$ by $3$ .

$4^{x - 1} = 8$

Step 7

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

$2^{2 (x - 1)} = 2^{3}$

Step 8

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

$2 (x - 1) = 3$

Step 9

Solve for $x$ .

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Step 9.1

Divide each term in $2 (x - 1) = 3$ by $2$ and simplify.

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Step 9.1.1

Divide each term in $2 (x - 1) = 3$ by $2$ .

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

Step 9.1.2

Simplify the left side.

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Step 9.1.2.1

Cancel the common factor of $2$ .

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Step 9.1.2.1.1

Cancel the common factor.

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

Step 9.1.2.1.2

Divide $x - 1$ by $1$ .

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

Step 9.2

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

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Step 9.2.1

Add $1$ to both sides of the equation.

$x = \frac{3}{2} + 1$

Step 9.2.2

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

$x = \frac{3}{2} + \frac{2}{2}$

Step 9.2.3

Combine the numerators over the common denominator.

$x = \frac{3 + 2}{2}$

Step 9.2.4

Add $3$ and $2$ .

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.5$

Mixed Number Form:

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