Finite Math Examples

$\frac{4 z^{6} + 7 x^{3} - 65}{z^{3}} = 0$

Step 1

Set the numerator equal to zero.

$4 z^{6} + 7 x^{3} - 65 = 0$

Step 2

Solve the equation for $z$ .

Tap for more steps...

Step 2.1

Move all terms not containing $z$ to the right side of the equation.

Tap for more steps...

Step 2.1.1

Subtract $7 x^{3}$ from both sides of the equation.

$4 z^{6} - 65 = - 7 x^{3}$

Step 2.1.2

Add $65$ to both sides of the equation.

$4 z^{6} = - 7 x^{3} + 65$

Step 2.2

Divide each term in $4 z^{6} = - 7 x^{3} + 65$ by $4$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $4 z^{6} = - 7 x^{3} + 65$ by $4$ .

$\frac{4 z^{6}}{4} = \frac{- 7 x^{3}}{4} + \frac{65}{4}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 z^{6}}{4} = \frac{- 7 x^{3}}{4} + \frac{65}{4}$

Step 2.2.2.1.2

Divide $z^{6}$ by $1$ .

$z^{6} = \frac{- 7 x^{3}}{4} + \frac{65}{4}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Move the negative in front of the fraction.

$z^{6} = - \frac{7 x^{3}}{4} + \frac{65}{4}$

Step 2.3

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

$z = \pm \sqrt[6]{- \frac{7 x^{3}}{4} + \frac{65}{4}}$

Step 2.4

Simplify $\pm \sqrt[6]{- \frac{7 x^{3}}{4} + \frac{65}{4}}$ .

Tap for more steps...

Step 2.4.1

Combine the numerators over the common denominator.

$z = \pm \sqrt[6]{\frac{- 7 x^{3} + 65}{4}}$

Step 2.4.2

Rewrite $\sqrt[6]{\frac{- 7 x^{3} + 65}{4}}$ as $\frac{\sqrt[6]{- 7 x^{3} + 65}}{\sqrt[6]{4}}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65}}{\sqrt[6]{4}}$

Step 2.4.3

Simplify the denominator.

Tap for more steps...

Step 2.4.3.1

Rewrite $4$ as $2^{2}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65}}{\sqrt[6]{2^{2}}}$

Step 2.4.3.2

Rewrite $\sqrt[6]{2^{2}}$ as $\sqrt[3]{\sqrt{2^{2}}}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65}}{\sqrt[3]{\sqrt{2^{2}}}}$

Step 2.4.3.3

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

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65}}{\sqrt[3]{2}}$

Step 2.4.4

Multiply $\frac{\sqrt[6]{- 7 x^{3} + 65}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 2.4.5

Combine and simplify the denominator.

Tap for more steps...

Step 2.4.5.1

Multiply $\frac{\sqrt[6]{- 7 x^{3} + 65}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 2.4.5.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 2.4.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 2.4.5.4

Add $1$ and $2$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 2.4.5.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

Tap for more steps...

Step 2.4.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 2.4.5.5.2

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

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 2.4.5.5.3

Combine $\frac{1}{3}$ and $3$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.4.5.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.4.5.5.4.1

Cancel the common factor.

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.4.5.5.4.2

Rewrite the expression.

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 2.4.5.5.5

Evaluate the exponent.

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} {\sqrt[3]{2}}^{2}}{2}$

Step 2.4.6

Simplify the numerator.

Tap for more steps...

Step 2.4.6.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} \sqrt[3]{2^{2}}}{2}$

Step 2.4.6.2

Raise $2$ to the power of $2$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} \sqrt[3]{4}}{2}$

Step 2.4.7

Simplify the numerator.

Tap for more steps...

Step 2.4.7.1

Rewrite the expression using the least common index of $6$ .

Tap for more steps...

Step 2.4.7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} \cdot 4^{\frac{1}{3}}}{2}$

Step 2.4.7.1.2

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{2}{6}}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} \cdot 4^{\frac{2}{6}}}{2}$

Step 2.4.7.1.3

Rewrite $4^{\frac{2}{6}}$ as $\sqrt[6]{4^{2}}$ .

$z = \pm \frac{\sqrt[6]{- 7 x^{3} + 65} \sqrt[6]{4^{2}}}{2}$

Step 2.4.7.2

Combine using the product rule for radicals.

$z = \pm \frac{\sqrt[6]{(- 7 x^{3} + 65) \cdot 4^{2}}}{2}$

Step 2.4.7.3

Raise $4$ to the power of $2$ .

$z = \pm \frac{\sqrt[6]{(- 7 x^{3} + 65) \cdot 16}}{2}$

Step 2.4.8

Reorder factors in $\pm \frac{\sqrt[6]{(- 7 x^{3} + 65) \cdot 16}}{2}$ .

$z = \pm \frac{\sqrt[6]{16 (- 7 x^{3} + 65)}}{2}$

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

$z = \frac{\sqrt[6]{16 (- 7 x^{3} + 65)}}{2}$

Step 2.5.2

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

$z = - \frac{\sqrt[6]{16 (- 7 x^{3} + 65)}}{2}$

Step 2.5.3

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

$z = \frac{\sqrt[6]{16 (- 7 x^{3} + 65)}}{2}$

$z = - \frac{\sqrt[6]{16 (- 7 x^{3} + 65)}}{2}$

$z = \frac{\sqrt[6]{16 (- 7 x^{3} + 65)}}{2}$

$z = - \frac{\sqrt[6]{16 (- 7 x^{3} + 65)}}{2}$

$z = \frac{\sqrt[6]{16 (- 7 x^{3} + 65)}}{2}$

$z = - \frac{\sqrt[6]{16 (- 7 x^{3} + 65)}}{2}$