Precalculus Examples

$7 x^{5} e^{- 3 x} - 8 x^{6} e^{- 3 x} = 0$

Step 1

Factor $x^{5} e^{- 3 x}$ out of $7 x^{5} e^{- 3 x} - 8 x^{6} e^{- 3 x}$ .

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Step 1.1

Factor $x^{5} e^{- 3 x}$ out of $7 x^{5} e^{- 3 x}$ .

$x^{5} e^{- 3 x} (7) - 8 x^{6} e^{- 3 x} = 0$

Step 1.2

Factor $x^{5} e^{- 3 x}$ out of $- 8 x^{6} e^{- 3 x}$ .

$x^{5} e^{- 3 x} (7) + x^{5} e^{- 3 x} (- 8 x) = 0$

Step 1.3

Factor $x^{5} e^{- 3 x}$ out of $x^{5} e^{- 3 x} (7) + x^{5} e^{- 3 x} (- 8 x)$ .

$x^{5} e^{- 3 x} (7 - 8 x) = 0$

Step 2

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

$x^{5} = 0$

$e^{- 3 x} = 0$

$7 - 8 x = 0$

Step 3

Set $x^{5}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{5}$ equal to $0$ .

$x^{5} = 0$

Step 3.2

Solve $x^{5} = 0$ for $x$ .

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Step 3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[5]{0}$

Step 3.2.2

Simplify $\sqrt[5]{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{5}$ .

$x = \sqrt[5]{0^{5}}$

Step 3.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 4

Set $e^{- 3 x}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $e^{- 3 x}$ equal to $0$ .

$e^{- 3 x} = 0$

Step 4.2

Solve $e^{- 3 x} = 0$ for $x$ .

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Step 4.2.1

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

$\ln (e^{- 3 x}) = \ln (0)$

Step 4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 4.2.3

There is no solution for $e^{- 3 x} = 0$

No solution

Step 5

Set $7 - 8 x$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $7 - 8 x$ equal to $0$ .

$7 - 8 x = 0$

Step 5.2

Solve $7 - 8 x = 0$ for $x$ .

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Step 5.2.1

Subtract $7$ from both sides of the equation.

$- 8 x = - 7$

Step 5.2.2

Divide each term in $- 8 x = - 7$ by $- 8$ and simplify.

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Step 5.2.2.1

Divide each term in $- 8 x = - 7$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 7}{- 8}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $- 8$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 7}{- 8}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{- 8}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{7}{8}$

Step 6

The final solution is all the values that make $x^{5} e^{- 3 x} (7 - 8 x) = 0$ true.

$x = 0, \frac{7}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{7}{8}$

Decimal Form:

$x = 0, 0.875$