Calculus Examples

$4^{3 x - 2} = 32^{\frac{3}{4} x - 1}$

Step 1

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

$2^{2 (3 x - 2)} = 2^{5 (\frac{3}{4} x - 1)}$

Step 2

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

$2 (3 x - 2) = 5 (\frac{3}{4} x - 1)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (3 x - 2)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 2 (3 x - 2) = 5 (\frac{3}{4} x - 1)$

Step 3.1.2

Simplify by adding zeros.

$2 (3 x - 2) = 5 (\frac{3}{4} x - 1)$

Step 3.1.3

Apply the distributive property.

$2 (3 x) + 2 \cdot - 2 = 5 (\frac{3}{4} x - 1)$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $3$ by $2$ .

$6 x + 2 \cdot - 2 = 5 (\frac{3}{4} x - 1)$

Step 3.1.4.2

Multiply $2$ by $- 2$ .

$6 x - 4 = 5 (\frac{3}{4} x - 1)$

Step 3.2

Simplify $5 (\frac{3}{4} x - 1)$ .

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Step 3.2.1

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

$6 x - 4 = 5 (\frac{3 x}{4} - 1)$

Step 3.2.2

Apply the distributive property.

$6 x - 4 = 5 \frac{3 x}{4} + 5 \cdot - 1$

Step 3.2.3

Multiply $5 \frac{3 x}{4}$ .

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Step 3.2.3.1

Combine $5$ and $\frac{3 x}{4}$ .

$6 x - 4 = \frac{5 (3 x)}{4} + 5 \cdot - 1$

Step 3.2.3.2

Multiply $3$ by $5$ .

$6 x - 4 = \frac{15 x}{4} + 5 \cdot - 1$

Step 3.2.4

Multiply $5$ by $- 1$ .

$6 x - 4 = \frac{15 x}{4} - 5$

Step 3.3

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

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Step 3.3.1

Subtract $\frac{15 x}{4}$ from both sides of the equation.

$6 x - 4 - \frac{15 x}{4} = - 5$

Step 3.3.2

To write $6 x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$6 x \cdot \frac{4}{4} - \frac{15 x}{4} - 4 = - 5$

Step 3.3.3

Combine $6 x$ and $\frac{4}{4}$ .

$\frac{6 x \cdot 4}{4} - \frac{15 x}{4} - 4 = - 5$

Step 3.3.4

Combine the numerators over the common denominator.

$\frac{6 x \cdot 4 - 15 x}{4} - 4 = - 5$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Factor $3 x$ out of $6 x \cdot 4 - 15 x$ .

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Step 3.3.5.1.1

Factor $3 x$ out of $6 x \cdot 4$ .

$\frac{3 x (2 \cdot 4) - 15 x}{4} - 4 = - 5$

Step 3.3.5.1.2

Factor $3 x$ out of $- 15 x$ .

$\frac{3 x (2 \cdot 4) + 3 x (- 5)}{4} - 4 = - 5$

Step 3.3.5.1.3

Factor $3 x$ out of $3 x (2 \cdot 4) + 3 x (- 5)$ .

$\frac{3 x (2 \cdot 4 - 5)}{4} - 4 = - 5$

Step 3.3.5.2

Multiply $2$ by $4$ .

$\frac{3 x (8 - 5)}{4} - 4 = - 5$

Step 3.3.5.3

Subtract $5$ from $8$ .

$\frac{3 x \cdot 3}{4} - 4 = - 5$

Step 3.3.5.4

Multiply $3$ by $3$ .

$\frac{9 x}{4} - 4 = - 5$

Step 3.4

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

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Step 3.4.1

Add $4$ to both sides of the equation.

$\frac{9 x}{4} = - 5 + 4$

Step 3.4.2

Add $- 5$ and $4$ .

$\frac{9 x}{4} = - 1$

Step 3.5

Multiply both sides of the equation by $\frac{4}{9}$ .

$\frac{4}{9} \cdot \frac{9 x}{4} = \frac{4}{9} \cdot - 1$

Step 3.6

Simplify both sides of the equation.

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Step 3.6.1

Simplify the left side.

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Step 3.6.1.1

Simplify $\frac{4}{9} \cdot \frac{9 x}{4}$ .

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Step 3.6.1.1.1

Cancel the common factor of $4$ .

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Step 3.6.1.1.1.1

Cancel the common factor.

$\frac{4}{9} \cdot \frac{9 x}{4} = \frac{4}{9} \cdot - 1$

Step 3.6.1.1.1.2

Rewrite the expression.

$\frac{1}{9} (9 x) = \frac{4}{9} \cdot - 1$

Step 3.6.1.1.2

Cancel the common factor of $9$ .

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Step 3.6.1.1.2.1

Factor $9$ out of $9 x$ .

$\frac{1}{9} (9 (x)) = \frac{4}{9} \cdot - 1$

Step 3.6.1.1.2.2

Cancel the common factor.

$\frac{1}{9} (9 x) = \frac{4}{9} \cdot - 1$

Step 3.6.1.1.2.3

Rewrite the expression.

$x = \frac{4}{9} \cdot - 1$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $\frac{4}{9} \cdot - 1$ .

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Step 3.6.2.1.1

Multiply $\frac{4}{9} \cdot - 1$ .

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Step 3.6.2.1.1.1

Combine $\frac{4}{9}$ and $- 1$ .

$x = \frac{4 \cdot - 1}{9}$

Step 3.6.2.1.1.2

Multiply $4$ by $- 1$ .

$x = \frac{- 4}{9}$

Step 3.6.2.1.2

Move the negative in front of the fraction.

$x = - \frac{4}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4}{9}$

Decimal Form:

$x = - 0.\overline{4}$