Precalculus Examples

$e^{x} (8 - e^{x}) = 16$

Step 1

Simplify $e^{x} (8 - e^{x})$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$e^{x} \cdot 8 + e^{x} (- e^{x}) = 16$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Move $8$ to the left of $e^{x}$ .

$8 \cdot e^{x} + e^{x} (- e^{x}) = 16$

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

$8 \cdot e^{x} - e^{x} e^{x} = 16$

Step 1.2

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

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Step 1.2.1

Move $e^{x}$ .

$8 e^{x} - (e^{x} e^{x}) = 16$

Step 1.2.2

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

$8 e^{x} - e^{x + x} = 16$

Step 1.2.3

Add $x$ and $x$ .

$8 e^{x} - e^{2 x} = 16$

Step 2

Rewrite $e^{2 x}$ as exponentiation.

$8 e^{x} - {(e^{x})}^{2} = 16$

Step 3

Substitute $u$ for $e^{x}$ .

$8 u - u^{2} = 16$

Step 4

Reorder $8 u$ and $- u^{2}$ .

$- u^{2} + 8 u = 16$

Step 5

Solve for $u$ .

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Step 5.1

Subtract $16$ from both sides of the equation.

$- u^{2} + 8 u - 16 = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor $- 1$ out of $- u^{2} + 8 u - 16$ .

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Step 5.2.1.1

Factor $- 1$ out of $- u^{2}$ .

$- (u^{2}) + 8 u - 16 = 0$

Step 5.2.1.2

Factor $- 1$ out of $8 u$ .

$- (u^{2}) - (- 8 u) - 16 = 0$

Step 5.2.1.3

Rewrite $- 16$ as $- 1 (16)$ .

$- (u^{2}) - (- 8 u) - 1 \cdot 16 = 0$

Step 5.2.1.4

Factor $- 1$ out of $- (u^{2}) - (- 8 u)$ .

$- (u^{2} - 8 u) - 1 \cdot 16 = 0$

Step 5.2.1.5

Factor $- 1$ out of $- (u^{2} - 8 u) - 1 (16)$ .

$- (u^{2} - 8 u + 16) = 0$

Step 5.2.2

Factor using the perfect square rule.

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Step 5.2.2.1

Rewrite $16$ as $4^{2}$ .

$- (u^{2} - 8 u + 4^{2}) = 0$

Step 5.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 u = 2 \cdot u \cdot 4$

Step 5.2.2.3

Rewrite the polynomial.

$- (u^{2} - 2 \cdot u \cdot 4 + 4^{2}) = 0$

Step 5.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 4$ .

$- {(u - 4)}^{2} = 0$

Step 5.3

Divide each term in $- {(u - 4)}^{2} = 0$ by $- 1$ and simplify.

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Step 5.3.1

Divide each term in $- {(u - 4)}^{2} = 0$ by $- 1$ .

$\frac{- {(u - 4)}^{2}}{- 1} = \frac{0}{- 1}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Dividing two negative values results in a positive value.

$\frac{{(u - 4)}^{2}}{1} = \frac{0}{- 1}$

Step 5.3.2.2

Divide ${(u - 4)}^{2}$ by $1$ .

${(u - 4)}^{2} = \frac{0}{- 1}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $0$ by $- 1$ .

${(u - 4)}^{2} = 0$

Step 5.4

Set the $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 5.5

Add $4$ to both sides of the equation.

$u = 4$

Step 6

Substitute $4$ for $u$ in $u = e^{x}$ .

$4 = e^{x}$

Step 7

Solve $4 = e^{x}$ .

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Step 7.1

Rewrite the equation as $e^{x} = 4$ .

$e^{x} = 4$

Step 7.2

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

$\ln (e^{x}) = \ln (4)$

Step 7.3

Expand the left side.

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Step 7.3.1

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

$x \ln (e) = \ln (4)$

Step 7.3.2

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

$x \cdot 1 = \ln (4)$

Step 7.3.3

Multiply $x$ by $1$ .

$x = \ln (4)$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \ln (4)$

Decimal Form:

$x = 1.38629436 \dots$