Finite Math Examples

$\frac{x^{3} + \sqrt{3 x - 8}}{5 - \sqrt[7]{x + 2}}$

Step 1

Set the radicand in $\sqrt{3 x - 8}$ greater than or equal to $0$ to find where the expression is defined.

$3 x - 8 \geq 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Add $8$ to both sides of the inequality.

$3 x \geq 8$

Step 2.2

Divide each term in $3 x \geq 8$ by $3$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $3 x \geq 8$ by $3$ .

$\frac{3 x}{3} \geq \frac{8}{3}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{8}{3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{8}{3}$

Step 3

Set the denominator in $\frac{x^{3} + \sqrt{3 x - 8}}{5 - \sqrt[7]{x + 2}}$ equal to $0$ to find where the expression is undefined.

$5 - \sqrt[7]{x + 2} = 0$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Subtract $5$ from both sides of the equation.

$- \sqrt[7]{x + 2} = - 5$

Step 4.2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $7$ .

${(- \sqrt[7]{x + 2})}^{7} = {(- 5)}^{7}$

Step 4.3

Simplify each side of the equation.

Tap for more steps...

Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x + 2}$ as ${(x + 2)}^{\frac{1}{7}}$ .

${(- {(x + 2)}^{\frac{1}{7}})}^{7} = {(- 5)}^{7}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Simplify ${(- {(x + 2)}^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 4.3.2.1.1

Apply the product rule to $- {(x + 2)}^{\frac{1}{7}}$ .

${(- 1)}^{7} {({(x + 2)}^{\frac{1}{7}})}^{7} = {(- 5)}^{7}$

Step 4.3.2.1.2

Raise $- 1$ to the power of $7$ .

$- {({(x + 2)}^{\frac{1}{7}})}^{7} = {(- 5)}^{7}$

Step 4.3.2.1.3

Multiply the exponents in ${({(x + 2)}^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 4.3.2.1.3.1

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

$- {(x + 2)}^{\frac{1}{7} \cdot 7} = {(- 5)}^{7}$

Step 4.3.2.1.3.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.3.2.1.3.2.1

Cancel the common factor.

$- {(x + 2)}^{\frac{1}{7} \cdot 7} = {(- 5)}^{7}$

Step 4.3.2.1.3.2.2

Rewrite the expression.

$- {(x + 2)}^{1} = {(- 5)}^{7}$

Step 4.3.2.1.4

Simplify.

$- (x + 2) = {(- 5)}^{7}$

Step 4.3.2.1.5

Apply the distributive property.

$- x - 1 \cdot 2 = {(- 5)}^{7}$

Step 4.3.2.1.6

Multiply $- 1$ by $2$ .

$- x - 2 = {(- 5)}^{7}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Raise $- 5$ to the power of $7$ .

$- x - 2 = - 78125$

Step 4.4

Solve for $x$ .

Tap for more steps...

Step 4.4.1

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

Tap for more steps...

Step 4.4.1.1

Add $2$ to both sides of the equation.

$- x = - 78125 + 2$

Step 4.4.1.2

Add $- 78125$ and $2$ .

$- x = - 78123$

Step 4.4.2

Divide each term in $- x = - 78123$ by $- 1$ and simplify.

Tap for more steps...

Step 4.4.2.1

Divide each term in $- x = - 78123$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 78123}{- 1}$

Step 4.4.2.2

Simplify the left side.

Tap for more steps...

Step 4.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 78123}{- 1}$

Step 4.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 78123}{- 1}$

Step 4.4.2.3

Simplify the right side.

Tap for more steps...

Step 4.4.2.3.1

Divide $- 78123$ by $- 1$ .

$x = 78123$

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[\frac{8}{3}, 78123) \cup (78123, \infty)$

Set-Builder Notation:

${x | x \geq \frac{8}{3}, x \neq 78123}$

Step 6