Algebra Examples

${(2^{x^{2} - 2 x})}^{4 - x} = 1$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({(2^{x^{2} - 2 x})}^{4 - x}) = \ln (1)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(2^{x^{2} - 2 x})}^{4 - x})$ by moving $4 - x$ outside the logarithm.

$(4 - x) \ln (2^{x^{2} - 2 x}) = \ln (1)$

Step 2.2

Expand $\ln (2^{x^{2} - 2 x})$ by moving $x^{2} - 2 x$ outside the logarithm.

$(4 - x) ((x^{2} - 2 x) \ln (2)) = \ln (1)$

Step 2.3

Remove parentheses.

$(4 - x) (x^{2} - 2 x) \ln (2) = \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $(4 - x) (x^{2} - 2 x) \ln (2)$ .

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Step 3.1.1

Expand $(4 - x) (x^{2} - 2 x)$ using the FOIL Method.

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Step 3.1.1.1

Apply the distributive property.

$(4 (x^{2} - 2 x) - x (x^{2} - 2 x)) \ln (2) = \ln (1)$

Step 3.1.1.2

Apply the distributive property.

$(4 x^{2} + 4 (- 2 x) - x (x^{2} - 2 x)) \ln (2) = \ln (1)$

Step 3.1.1.3

Apply the distributive property.

$(4 x^{2} + 4 (- 2 x) - x \cdot x^{2} - x (- 2 x)) \ln (2) = \ln (1)$

Step 3.1.2

Simplify and combine like terms.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply $- 2$ by $4$ .

$(4 x^{2} - 8 x - x \cdot x^{2} - x (- 2 x)) \ln (2) = \ln (1)$

Step 3.1.2.1.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.1.2.1.2.1

Move $x^{2}$ .

$(4 x^{2} - 8 x - (x^{2} x) - x (- 2 x)) \ln (2) = \ln (1)$

Step 3.1.2.1.2.2

Multiply $x^{2}$ by $x$ .

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Step 3.1.2.1.2.2.1

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

$(4 x^{2} - 8 x - (x^{2} x^{1}) - x (- 2 x)) \ln (2) = \ln (1)$

Step 3.1.2.1.2.2.2

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

$(4 x^{2} - 8 x - x^{2 + 1} - x (- 2 x)) \ln (2) = \ln (1)$

Step 3.1.2.1.2.3

Add $2$ and $1$ .

$(4 x^{2} - 8 x - x^{3} - x (- 2 x)) \ln (2) = \ln (1)$

Step 3.1.2.1.3

Rewrite using the commutative property of multiplication.

$(4 x^{2} - 8 x - x^{3} - 1 \cdot - 2 x \cdot x) \ln (2) = \ln (1)$

Step 3.1.2.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.2.1.4.1

Move $x$ .

$(4 x^{2} - 8 x - x^{3} - 1 \cdot - 2 (x \cdot x)) \ln (2) = \ln (1)$

Step 3.1.2.1.4.2

Multiply $x$ by $x$ .

$(4 x^{2} - 8 x - x^{3} - 1 \cdot - 2 x^{2}) \ln (2) = \ln (1)$

Step 3.1.2.1.5

Multiply $- 1$ by $- 2$ .

$(4 x^{2} - 8 x - x^{3} + 2 x^{2}) \ln (2) = \ln (1)$

Step 3.1.2.2

Add $4 x^{2}$ and $2 x^{2}$ .

$(6 x^{2} - 8 x - x^{3}) \ln (2) = \ln (1)$

Step 3.1.3

Apply the distributive property.

$6 x^{2} \ln (2) - 8 x \ln (2) - x^{3} \ln (2) = \ln (1)$

Step 4

Simplify the right side.

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Step 4.1

The natural logarithm of $1$ is $0$ .

$6 x^{2} \ln (2) - 8 x \ln (2) - x^{3} \ln (2) = 0$

Step 5

Simplify the left side.

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Step 5.1

Move $- 8 x \ln (2)$ .

$6 x^{2} \ln (2) - x^{3} \ln (2) - 8 x \ln (2) = 0$

Step 5.2

Reorder $6 x^{2} \ln (2)$ and $- x^{3} \ln (2)$ .

$- x^{3} \ln (2) + 6 x^{2} \ln (2) - 8 x \ln (2) = 0$

Step 6

Factor $x \ln (2)$ out of $- x^{3} \ln (2) + 6 x^{2} \ln (2) - 8 x \ln (2)$ .

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Step 6.1

Factor $x \ln (2)$ out of $- x^{3} \ln (2)$ .

$x \ln (2) (- x^{2}) + 6 x^{2} \ln (2) - 8 x \ln (2) = 0$

Step 6.2

Factor $x \ln (2)$ out of $6 x^{2} \ln (2)$ .

$x \ln (2) (- x^{2}) + x \ln (2) (6 x) - 8 x \ln (2) = 0$

Step 6.3

Factor $x \ln (2)$ out of $- 8 x \ln (2)$ .

$x \ln (2) (- x^{2}) + x \ln (2) (6 x) + x \ln (2) \cdot - 8 = 0$

Step 6.4

Factor $x \ln (2)$ out of $x \ln (2) (- x^{2}) + x \ln (2) (6 x)$ .

$x \ln (2) (- x^{2} + 6 x) + x \ln (2) \cdot - 8 = 0$

Step 6.5

Factor $x \ln (2)$ out of $x \ln (2) (- x^{2} + 6 x) + x \ln (2) \cdot - 8$ .

$x \ln (2) (- x^{2} + 6 x - 8) = 0$

Step 7

Factor.

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Step 7.1

Factor by grouping.

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Step 7.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 8 = 8$ and whose sum is $b = 6$ .

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Step 7.1.1.1

Factor $6$ out of $6 x$ .

$x \ln (2) (- x^{2} + 6 (x) - 8) = 0$

Step 7.1.1.2

Rewrite $6$ as $2$ plus $4$

$x \ln (2) (- x^{2} + (2 + 4) x - 8) = 0$

Step 7.1.1.3

Apply the distributive property.

$x \ln (2) (- x^{2} + 2 x + 4 x - 8) = 0$

Step 7.1.2

Factor out the greatest common factor from each group.

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Step 7.1.2.1

Group the first two terms and the last two terms.

$x \ln (2) ((- x^{2} + 2 x) + 4 x - 8) = 0$

Step 7.1.2.2

Factor out the greatest common factor (GCF) from each group.

$x \ln (2) (x (- x + 2) - 4 (- x + 2)) = 0$

Step 7.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$x \ln (2) ((- x + 2) (x - 4)) = 0$

Step 7.2

Remove unnecessary parentheses.

$x \ln (2) (- x + 2) (x - 4) = 0$

Step 8

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

$x = 0$

$- x + 2 = 0$

$x - 4 = 0$

Step 9

Set $x$ equal to $0$ .

$x = 0$

Step 10

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

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Step 10.1

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

$- x + 2 = 0$

Step 10.2

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

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Step 10.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 10.2.2

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

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Step 10.2.2.1

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

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

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 10.2.2.2.2

Divide $x$ by $1$ .

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

Step 10.2.2.3

Simplify the right side.

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Step 10.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 11

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 11.2

Add $4$ to both sides of the equation.

$x = 4$

Step 12

The final solution is all the values that make $x \ln (2) (- x + 2) (x - 4) = 0$ true.

$x = 0, 2, 4$