Calculus Examples

${(\frac{1}{8})}^{x - 1} = 2^{3 - 2 x^{2}}$

Step 1

Apply the product rule to $\frac{1}{8}$ .

$\frac{1^{x - 1}}{8^{x - 1}} = 2^{3 - 2 x^{2}}$

Step 2

One to any power is one.

$\frac{1}{8^{x - 1}} = 2^{3 - 2 x^{2}}$

Step 3

Move $8^{x - 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$8^{- (x - 1)} = 2^{3 - 2 x^{2}}$

Step 4

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

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

Step 5

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

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

Step 6

Solve for $x$ .

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Step 6.1

Simplify $3 (- (x - 1))$ .

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Step 6.1.1

Rewrite.

$0 + 0 + 3 (- (x - 1)) = 3 - 2 x^{2}$

Step 6.1.2

Simplify by adding zeros.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Multiply $- 1$ by $- 1$ .

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

Step 6.1.5

Apply the distributive property.

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

Step 6.1.6

Multiply.

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Step 6.1.6.1

Multiply $- 1$ by $3$ .

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

Step 6.1.6.2

Multiply $3$ by $1$ .

$- 3 x + 3 = 3 - 2 x^{2}$

Step 6.2

Add $2 x^{2}$ to both sides of the equation.

$- 3 x + 3 + 2 x^{2} = 3$

Step 6.3

Subtract $3$ from both sides of the equation.

$- 3 x + 3 + 2 x^{2} - 3 = 0$

Step 6.4

Combine the opposite terms in $- 3 x + 3 + 2 x^{2} - 3$ .

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Step 6.4.1

Subtract $3$ from $3$ .

$- 3 x + 2 x^{2} + 0 = 0$

Step 6.4.2

Add $- 3 x + 2 x^{2}$ and $0$ .

$- 3 x + 2 x^{2} = 0$

Step 6.5

Factor $- x$ out of $- 3 x + 2 x^{2}$ .

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Step 6.5.1

Reorder $- 3 x$ and $2 x^{2}$ .

$2 x^{2} - 3 x = 0$

Step 6.5.2

Factor $- x$ out of $2 x^{2}$ .

$- x (- 2 x) - 3 x = 0$

Step 6.5.3

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

$- x (- 2 x) - x \cdot 3 = 0$

Step 6.5.4

Factor $- x$ out of $- x (- 2 x) - x (3)$ .

$- x (- 2 x + 3) = 0$

Step 6.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$- 2 x + 3 = 0$

Step 6.7

Set $x$ equal to $0$ .

$x = 0$

Step 6.8

Set $- 2 x + 3$ equal to $0$ and solve for $x$ .

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Step 6.8.1

Set $- 2 x + 3$ equal to $0$ .

$- 2 x + 3 = 0$

Step 6.8.2

Solve $- 2 x + 3 = 0$ for $x$ .

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Step 6.8.2.1

Subtract $3$ from both sides of the equation.

$- 2 x = - 3$

Step 6.8.2.2

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

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Step 6.8.2.2.1

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

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

Step 6.8.2.2.2

Simplify the left side.

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Step 6.8.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 6.8.2.2.2.1.1

Cancel the common factor.

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

Step 6.8.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.8.2.2.3

Simplify the right side.

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Step 6.8.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 6.9

The final solution is all the values that make $- x (- 2 x + 3) = 0$ true.

$x = 0, \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{3}{2}$

Decimal Form:

$x = 0, 1.5$

Mixed Number Form:

$x = 0, 1 \frac{1}{2}$