Calculus Examples

$- \frac{8}{11} \cdot (t e^{- \frac{t}{11}}) + 8 e^{- \frac{t}{11}} = 0$

Step 1

Simplify $- \frac{8}{11} \cdot (t e^{- \frac{t}{11}}) + 8 e^{- \frac{t}{11}}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply $- \frac{8}{11} (t e^{- \frac{t}{11}})$ .

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Step 1.1.1.1

Combine $t$ and $\frac{8}{11}$ .

$- \frac{t \cdot 8}{11} e^{- \frac{t}{11}} + 8 e^{- \frac{t}{11}} = 0$

Step 1.1.1.2

Combine $e^{- \frac{t}{11}}$ and $\frac{t \cdot 8}{11}$ .

$- \frac{e^{- \frac{t}{11}} (t \cdot 8)}{11} + 8 e^{- \frac{t}{11}} = 0$

Step 1.1.2

Remove unnecessary parentheses.

$- \frac{e^{- \frac{t}{11}} t \cdot 8}{11} + 8 e^{- \frac{t}{11}} = 0$

Step 1.1.3

Move $8$ to the left of $e^{- \frac{t}{11}} t$ .

$- \frac{8 e^{- \frac{t}{11}} t}{11} + 8 e^{- \frac{t}{11}} = 0$

Step 1.2

Reorder factors in $- \frac{8 e^{- \frac{t}{11}} t}{11} + 8 e^{- \frac{t}{11}}$ .

$- \frac{8 t e^{- \frac{t}{11}}}{11} + 8 e^{- \frac{t}{11}} = 0$

Step 2

Factor $8 e^{- \frac{t}{11}}$ out of $- \frac{8 t e^{- \frac{t}{11}}}{11} + 8 e^{- \frac{t}{11}}$ .

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Step 2.1

Factor $8 e^{- \frac{t}{11}}$ out of $- \frac{8 t e^{- \frac{t}{11}}}{11}$ .

$8 e^{- \frac{t}{11}} (- \frac{t}{11}) + 8 e^{- \frac{t}{11}} = 0$

Step 2.2

Factor $8 e^{- \frac{t}{11}}$ out of $8 e^{- \frac{t}{11}}$ .

$8 e^{- \frac{t}{11}} (- \frac{t}{11}) + 8 e^{- \frac{t}{11}} (1) = 0$

Step 2.3

Factor $8 e^{- \frac{t}{11}}$ out of $8 e^{- \frac{t}{11}} (- \frac{t}{11}) + 8 e^{- \frac{t}{11}} (1)$ .

$8 e^{- \frac{t}{11}} (- \frac{t}{11} + 1) = 0$

Step 3

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

$e^{- \frac{t}{11}} = 0$

$- \frac{t}{11} + 1 = 0$

Step 4

Set $e^{- \frac{t}{11}}$ equal to $0$ and solve for $t$ .

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Step 4.1

Set $e^{- \frac{t}{11}}$ equal to $0$ .

$e^{- \frac{t}{11}} = 0$

Step 4.2

Solve $e^{- \frac{t}{11}} = 0$ for $t$ .

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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.

$\ln (e^{- \frac{t}{11}}) = \ln (0)$

Step 4.2.2

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

Undefined

Step 4.2.3

There is no solution for $e^{- \frac{t}{11}} = 0$

No solution

Step 5

Set $- \frac{t}{11} + 1$ equal to $0$ and solve for $t$ .

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Step 5.1

Set $- \frac{t}{11} + 1$ equal to $0$ .

$- \frac{t}{11} + 1 = 0$

Step 5.2

Solve $- \frac{t}{11} + 1 = 0$ for $t$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$- \frac{t}{11} = - 1$

Step 5.2.2

Multiply both sides of the equation by $- 11$ .

$- 11 (- \frac{t}{11}) = - 11 \cdot - 1$

Step 5.2.3

Simplify both sides of the equation.

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Step 5.2.3.1

Simplify the left side.

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Step 5.2.3.1.1

Simplify $- 11 (- \frac{t}{11})$ .

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Step 5.2.3.1.1.1

Cancel the common factor of $11$ .

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Step 5.2.3.1.1.1.1

Move the leading negative in $- \frac{t}{11}$ into the numerator.

$- 11 \frac{- t}{11} = - 11 \cdot - 1$

Step 5.2.3.1.1.1.2

Factor $11$ out of $- 11$ .

$11 (- 1) \frac{- t}{11} = - 11 \cdot - 1$

Step 5.2.3.1.1.1.3

Cancel the common factor.

$11 \cdot - 1 \frac{- t}{11} = - 11 \cdot - 1$

Step 5.2.3.1.1.1.4

Rewrite the expression.

$- - t = - 11 \cdot - 1$

Step 5.2.3.1.1.2

Multiply.

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Step 5.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 t = - 11 \cdot - 1$

Step 5.2.3.1.1.2.2

Multiply $t$ by $1$ .

$t = - 11 \cdot - 1$

Step 5.2.3.2

Simplify the right side.

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Step 5.2.3.2.1

Multiply $- 11$ by $- 1$ .

$t = 11$

Step 6

The final solution is all the values that make $8 e^{- \frac{t}{11}} (- \frac{t}{11} + 1) = 0$ true.

$t = 11$