Precalculus Examples

$x^{7} (8 e^{8 x}) + 7 x^{6} e^{8 x}$

Step 1

Set $x^{7} (8 e^{8 x}) + 7 x^{6} e^{8 x}$ equal to $0$ .

$x^{7} (8 e^{8 x}) + 7 x^{6} e^{8 x} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite using the commutative property of multiplication.

$8 x^{7} e^{8 x} + 7 x^{6} e^{8 x} = 0$

Step 2.2

Factor $x^{6} e^{8 x}$ out of $8 x^{7} e^{8 x} + 7 x^{6} e^{8 x}$ .

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Step 2.2.1

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

$x^{6} e^{8 x} (8 x) + 7 x^{6} e^{8 x} = 0$

Step 2.2.2

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

$x^{6} e^{8 x} (8 x) + x^{6} e^{8 x} (7) = 0$

Step 2.2.3

Factor $x^{6} e^{8 x}$ out of $x^{6} e^{8 x} (8 x) + x^{6} e^{8 x} (7)$ .

$x^{6} e^{8 x} (8 x + 7) = 0$

Step 2.3

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

$x^{6} = 0$

$e^{8 x} = 0$

$8 x + 7 = 0$

Step 2.4

Set $x^{6}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{6}$ equal to $0$ .

$x^{6} = 0$

Step 2.4.2

Solve $x^{6} = 0$ for $x$ .

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Step 2.4.2.1

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

$x = \pm \sqrt[6]{0}$

Step 2.4.2.2

Simplify $\pm \sqrt[6]{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{6}$ .

$x = \pm \sqrt[6]{0^{6}}$

Step 2.4.2.2.2

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

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $e^{8 x}$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $e^{8 x}$ equal to $0$ .

$e^{8 x} = 0$

Step 2.5.2

Solve $e^{8 x} = 0$ for $x$ .

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Step 2.5.2.1

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

$\ln (e^{8 x}) = \ln (0)$

Step 2.5.2.2

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

Undefined

Step 2.5.2.3

There is no solution for $e^{8 x} = 0$

No solution

Step 2.6

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

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Step 2.6.1

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

$8 x + 7 = 0$

Step 2.6.2

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

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Step 2.6.2.1

Subtract $7$ from both sides of the equation.

$8 x = - 7$

Step 2.6.2.2

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

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Step 2.6.2.2.1

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

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

Step 2.6.2.2.2

Simplify the left side.

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Step 2.6.2.2.2.1

Cancel the common factor of $8$ .

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Step 2.6.2.2.2.1.1

Cancel the common factor.

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

Step 2.6.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.6.2.2.3

Simplify the right side.

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Step 2.6.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.7

The final solution is all the values that make $x^{6} e^{8 x} (8 x + 7) = 0$ true.

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

Step 3