Calculus Examples

$1 - \frac{1}{3 t^{\frac{2}{3}}}$

Step 1

Set $1 - \frac{1}{3 t^{\frac{2}{3}}}$ equal to $0$ .

$1 - \frac{1}{3 t^{\frac{2}{3}}} = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $1$ from both sides of the equation.

$- \frac{1}{3 t^{\frac{2}{3}}} = - 1$

Step 2.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 t^{\frac{2}{3}}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$3 t^{\frac{2}{3}}$

Step 2.3

Multiply each term in $- \frac{1}{3 t^{\frac{2}{3}}} = - 1$ by $3 t^{\frac{2}{3}}$ to eliminate the fractions.

Tap for more steps...

Step 2.3.1

Multiply each term in $- \frac{1}{3 t^{\frac{2}{3}}} = - 1$ by $3 t^{\frac{2}{3}}$ .

$- \frac{1}{3 t^{\frac{2}{3}}} (3 t^{\frac{2}{3}}) = - (3 t^{\frac{2}{3}})$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $3 t^{\frac{2}{3}}$ .

Tap for more steps...

Step 2.3.2.1.1

Move the leading negative in $- \frac{1}{3 t^{\frac{2}{3}}}$ into the numerator.

$\frac{- 1}{3 t^{\frac{2}{3}}} (3 t^{\frac{2}{3}}) = - (3 t^{\frac{2}{3}})$

Step 2.3.2.1.2

Cancel the common factor.

$\frac{- 1}{3 t^{\frac{2}{3}}} (3 t^{\frac{2}{3}}) = - (3 t^{\frac{2}{3}})$

Step 2.3.2.1.3

Rewrite the expression.

$- 1 = - (3 t^{\frac{2}{3}})$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Multiply $3$ by $- 1$ .

$- 1 = - 3 t^{\frac{2}{3}}$

Step 2.4

Solve the equation.

Tap for more steps...

Step 2.4.1

Rewrite the equation as $- 3 t^{\frac{2}{3}} = - 1$ .

$- 3 t^{\frac{2}{3}} = - 1$

Step 2.4.2

Divide each term in $- 3 t^{\frac{2}{3}} = - 1$ by $- 3$ and simplify.

Tap for more steps...

Step 2.4.2.1

Divide each term in $- 3 t^{\frac{2}{3}} = - 1$ by $- 3$ .

$\frac{- 3 t^{\frac{2}{3}}}{- 3} = \frac{- 1}{- 3}$

Step 2.4.2.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.2.1

Cancel the common factor.

$\frac{- 3 t^{\frac{2}{3}}}{- 3} = \frac{- 1}{- 3}$

Step 2.4.2.2.2

Divide $t^{\frac{2}{3}}$ by $1$ .

$t^{\frac{2}{3}} = \frac{- 1}{- 3}$

Step 2.4.2.3

Simplify the right side.

Tap for more steps...

Step 2.4.2.3.1

Dividing two negative values results in a positive value.

$t^{\frac{2}{3}} = \frac{1}{3}$

Step 2.4.3

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(t^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(\frac{1}{3})}^{\frac{3}{2}}$

Step 2.4.4

Simplify the exponent.

Tap for more steps...

Step 2.4.4.1

Simplify the left side.

Tap for more steps...

Step 2.4.4.1.1

Simplify ${(t^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 2.4.4.1.1.1

Multiply the exponents in ${(t^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 2.4.4.1.1.1.1

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

$t^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{1}{3})}^{\frac{3}{2}}$

Step 2.4.4.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.4.1.1.1.2.1

Cancel the common factor.

$t^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{1}{3})}^{\frac{3}{2}}$

Step 2.4.4.1.1.1.2.2

Rewrite the expression.

$t^{\frac{1}{3} \cdot 3} = \pm {(\frac{1}{3})}^{\frac{3}{2}}$

Step 2.4.4.1.1.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.4.4.1.1.1.3.1

Cancel the common factor.

$t^{\frac{1}{3} \cdot 3} = \pm {(\frac{1}{3})}^{\frac{3}{2}}$

Step 2.4.4.1.1.1.3.2

Rewrite the expression.

$t^{1} = \pm {(\frac{1}{3})}^{\frac{3}{2}}$

Step 2.4.4.1.1.2

Simplify.

$t = \pm {(\frac{1}{3})}^{\frac{3}{2}}$

Step 2.4.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.4.2.1

Simplify $\pm {(\frac{1}{3})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 2.4.4.2.1.1

Apply the product rule to $\frac{1}{3}$ .

$t = \pm \frac{1^{\frac{3}{2}}}{3^{\frac{3}{2}}}$

Step 2.4.4.2.1.2

One to any power is one.

$t = \pm \frac{1}{3^{\frac{3}{2}}}$

Step 2.4.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$t = \frac{1}{3^{\frac{3}{2}}}$

Step 2.4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \frac{1}{3^{\frac{3}{2}}}$

Step 2.4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = \frac{1}{3^{\frac{3}{2}}}, - \frac{1}{3^{\frac{3}{2}}}$

Step 3