Calculus Examples

$0 = \frac{35}{2} + \frac{13 \sqrt[30]{e^{29 t}}}{2 e^{t}}$

Step 1

Rewrite the equation as $\frac{35}{2} + \frac{13 \sqrt[30]{e^{29 t}}}{2 e^{t}} = 0$ .

$\frac{35}{2} + \frac{13 \sqrt[30]{e^{29 t}}}{2 e^{t}} = 0$

Step 2

Solve for $13 \sqrt[30]{e^{29 t}}$ .

Tap for more steps...

Step 2.1

Subtract $\frac{35}{2}$ from both sides of the equation.

$\frac{13 \sqrt[30]{e^{29 t}}}{2 e^{t}} = - \frac{35}{2}$

Step 2.2

Multiply both sides by $2 e^{t}$ .

$\frac{13 \sqrt[30]{e^{29 t}}}{2 e^{t}} (2 e^{t}) = - \frac{35}{2} (2 e^{t})$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Simplify the left side.

Tap for more steps...

Step 2.3.1.1

Simplify $\frac{13 \sqrt[30]{e^{29 t}}}{2 e^{t}} (2 e^{t})$ .

Tap for more steps...

Step 2.3.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{13 \sqrt[30]{e^{29 t}}}{2 e^{t}} e^{t} = - \frac{35}{2} (2 e^{t})$

Step 2.3.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.1.1.2.1

Factor $2$ out of $2 e^{t}$ .

$2 \frac{13 \sqrt[30]{e^{29 t}}}{2 (e^{t})} e^{t} = - \frac{35}{2} (2 e^{t})$

Step 2.3.1.1.2.2

Cancel the common factor.

$2 \frac{13 \sqrt[30]{e^{29 t}}}{2 e^{t}} e^{t} = - \frac{35}{2} (2 e^{t})$

Step 2.3.1.1.2.3

Rewrite the expression.

$\frac{13 \sqrt[30]{e^{29 t}}}{e^{t}} e^{t} = - \frac{35}{2} (2 e^{t})$

Step 2.3.1.1.3

Cancel the common factor of $e^{t}$ .

Tap for more steps...

Step 2.3.1.1.3.1

Cancel the common factor.

$\frac{13 \sqrt[30]{e^{29 t}}}{e^{t}} e^{t} = - \frac{35}{2} (2 e^{t})$

Step 2.3.1.1.3.2

Rewrite the expression.

$13 \sqrt[30]{e^{29 t}} = - \frac{35}{2} (2 e^{t})$

Step 2.3.2

Simplify the right side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.1.1

Move the leading negative in $- \frac{35}{2}$ into the numerator.

$13 \sqrt[30]{e^{29 t}} = \frac{- 35}{2} (2 e^{t})$

Step 2.3.2.1.2

Factor $2$ out of $2 e^{t}$ .

$13 \sqrt[30]{e^{29 t}} = \frac{- 35}{2} (2 (e^{t}))$

Step 2.3.2.1.3

Cancel the common factor.

$13 \sqrt[30]{e^{29 t}} = \frac{- 35}{2} (2 e^{t})$

Step 2.3.2.1.4

Rewrite the expression.

$13 \sqrt[30]{e^{29 t}} = - 35 e^{t}$

Step 3

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $30$ .

${(13 \sqrt[30]{e^{29 t}})}^{30} = {(- 35 e^{t})}^{30}$

Step 4

Simplify each side of the equation.

Tap for more steps...

Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[30]{e^{29 t}}$ as $e^{\frac{29 t}{30}}$ .

${(13 e^{\frac{29 t}{30}})}^{30} = {(- 35 e^{t})}^{30}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Simplify ${(13 e^{\frac{29 t}{30}})}^{30}$ .

Tap for more steps...

Step 4.2.1.1

Apply the product rule to $13 e^{\frac{29 t}{30}}$ .

$13^{30} {(e^{\frac{29 t}{30}})}^{30} = {(- 35 e^{t})}^{30}$

Step 4.2.1.2

Multiply the exponents in ${(e^{\frac{29 t}{30}})}^{30}$ .

Tap for more steps...

Step 4.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$13^{30} e^{\frac{29 t}{30} \cdot 30} = {(- 35 e^{t})}^{30}$

Step 4.2.1.2.2

Cancel the common factor of $30$ .

Tap for more steps...

Step 4.2.1.2.2.1

Cancel the common factor.

$13^{30} e^{\frac{29 t}{30} \cdot 30} = {(- 35 e^{t})}^{30}$

Step 4.2.1.2.2.2

Rewrite the expression.

$13^{30} e^{29 t} = {(- 35 e^{t})}^{30}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Simplify ${(- 35 e^{t})}^{30}$ .

Tap for more steps...

Step 4.3.1.1

Apply the product rule to $- 35 e^{t}$ .

$13^{30} e^{29 t} = {(- 35)}^{30} {(e^{t})}^{30}$

Step 4.3.1.2

Multiply the exponents in ${(e^{t})}^{30}$ .

Tap for more steps...

Step 4.3.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$13^{30} e^{29 t} = {(- 35)}^{30} e^{t \cdot 30}$

Step 4.3.1.2.2

Move $30$ to the left of $t$ .

$13^{30} e^{29 t} = {(- 35)}^{30} e^{30 t}$

Step 5

Solve for $t$ .

Tap for more steps...

Step 5.1

Subtract ${(- 35)}^{30} e^{30 t}$ from both sides of the equation.

$13^{30} e^{29 t} - {(- 35)}^{30} e^{30 t} = 0$

Step 5.2

Rewrite $e^{29 t}$ as exponentiation.

$13^{30} {(e^{t})}^{29} - {(- 35)}^{30} e^{30 t} = 0$

Step 5.3

Rewrite $e^{30 t}$ as exponentiation.

$13^{30} {(e^{t})}^{29} - {(- 35)}^{30} {(e^{t})}^{30} = 0$

Step 5.4

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

$13^{30} u^{29} - {(- 35)}^{30} u^{30} = 0$

Step 5.5

Reorder $13^{30} u^{29}$ and $- {(- 35)}^{30} u^{30}$ .

$- {(- 35)}^{30} u^{30} + 13^{30} u^{29} = 0$

Step 5.6

Solve for $u$ .

Tap for more steps...

Step 5.6.1

Factor $- u^{29}$ out of $- {(- 35)}^{30} u^{30} + 13^{30} u^{29}$ .

Tap for more steps...

Step 5.6.1.1

Factor $- u^{29}$ out of $- {(- 35)}^{30} u^{30}$ .

$- u^{29} ({(- 35)}^{30} u) + 13^{30} u^{29} = 0$

Step 5.6.1.2

Factor $- u^{29}$ out of $13^{30} u^{29}$ .

$- u^{29} ({(- 35)}^{30} u) - u^{29} (- 13^{30}) = 0$

Step 5.6.1.3

Factor $- u^{29}$ out of $- u^{29} ({(- 35)}^{30} u) - u^{29} (- 13^{30})$ .

$- u^{29} ({(- 35)}^{30} u - 13^{30}) = 0$

Step 5.6.2

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

$u^{29} = 0$

${(- 35)}^{30} u - 13^{30} = 0$

Step 5.6.3

Set $u^{29}$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 5.6.3.1

Set $u^{29}$ equal to $0$ .

$u^{29} = 0$

Step 5.6.3.2

Solve $u^{29} = 0$ for $u$ .

Tap for more steps...

Step 5.6.3.2.1

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

$u = \sqrt[29]{0}$

Step 5.6.3.2.2

Simplify $\sqrt[29]{0}$ .

Tap for more steps...

Step 5.6.3.2.2.1

Rewrite $0$ as $0^{29}$ .

$u = \sqrt[29]{0^{29}}$

Step 5.6.3.2.2.2

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

$u = 0$

Step 5.6.4

Set ${(- 35)}^{30} u - 13^{30}$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 5.6.4.1

Set ${(- 35)}^{30} u - 13^{30}$ equal to $0$ .

${(- 35)}^{30} u - 13^{30} = 0$

Step 5.6.4.2

Solve ${(- 35)}^{30} u - 13^{30} = 0$ for $u$ .

Tap for more steps...

Step 5.6.4.2.1

Add $13^{30}$ to both sides of the equation.

${(- 35)}^{30} u = 13^{30}$

Step 5.6.4.2.2

Divide each term in ${(- 35)}^{30} u = 13^{30}$ by ${(- 35)}^{30}$ and simplify.

Tap for more steps...

Step 5.6.4.2.2.1

Divide each term in ${(- 35)}^{30} u = 13^{30}$ by ${(- 35)}^{30}$ .

$\frac{{(- 35)}^{30} u}{{(- 35)}^{30}} = \frac{13^{30}}{{(- 35)}^{30}}$

Step 5.6.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.6.4.2.2.2.1

Cancel the common factor of ${(- 35)}^{30}$ .

Tap for more steps...

Step 5.6.4.2.2.2.1.1

Cancel the common factor.

$\frac{{(- 35)}^{30} u}{{(- 35)}^{30}} = \frac{13^{30}}{{(- 35)}^{30}}$

Step 5.6.4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{13^{30}}{{(- 35)}^{30}}$

Step 5.6.5

The final solution is all the values that make $- u^{29} ({(- 35)}^{30} u - 13^{30}) = 0$ true.

$u = 0, \frac{13^{30}}{{(- 35)}^{30}}$

Step 5.7

Substitute $0$ for $u$ in $u = e^{t}$ .

$0 = e^{t}$

Step 5.8

Solve $0 = e^{t}$ .

Tap for more steps...

Step 5.8.1

Rewrite the equation as $e^{t} = 0$ .

$e^{t} = 0$

Step 5.8.2

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

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

Step 5.8.3

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

Undefined

Step 5.8.4

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

No solution

Step 5.9

Substitute $\frac{13^{30}}{{(- 35)}^{30}}$ for $u$ in $u = e^{t}$ .

$\frac{13^{30}}{{(- 35)}^{30}} = e^{t}$

Step 5.10

Solve $\frac{13^{30}}{{(- 35)}^{30}} = e^{t}$ .

Tap for more steps...

Step 5.10.1

Rewrite the equation as $e^{t} = \frac{13^{30}}{{(- 35)}^{30}}$ .

$e^{t} = \frac{13^{30}}{{(- 35)}^{30}}$

Step 5.10.2

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

$\ln (e^{t}) = \ln (\frac{13^{30}}{{(- 35)}^{30}})$

Step 5.10.3

Expand the left side.

Tap for more steps...

Step 5.10.3.1

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

$t \ln (e) = \ln (\frac{13^{30}}{{(- 35)}^{30}})$

Step 5.10.3.2

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

$t \cdot 1 = \ln (\frac{13^{30}}{{(- 35)}^{30}})$

Step 5.10.3.3

Multiply $t$ by $1$ .

$t = \ln (\frac{13^{30}}{{(- 35)}^{30}})$

Step 6

Exclude the solutions that do not make $0 = \frac{35}{2} + \frac{13 \sqrt[30]{e^{29 t}}}{2 e^{t}}$ true.

$No solution$