Trigonometry Examples

$\sqrt[4]{\frac{125}{x^{8}}}$

Step 1

Set the denominator in $\frac{125}{x^{8}}$ equal to $0$ to find where the expression is undefined.

$x^{8} = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

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

$x = \pm \sqrt[8]{0}$

Step 2.2

Simplify $\pm \sqrt[8]{0}$ .

Tap for more steps...

Step 2.2.1

Rewrite $0$ as $0^{8}$ .

$x = \pm \sqrt[8]{0^{8}}$

Step 2.2.2

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

$x = \pm 0$

Step 2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

Set the radicand in $\sqrt[4]{\frac{125}{x^{8}}}$ less than $0$ to find where the expression is undefined.

$\frac{125}{x^{8}} < 0$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{8} = 0$

Step 4.2

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

$x = \pm \sqrt[8]{0}$

Step 4.3

Simplify $\pm \sqrt[8]{0}$ .

Tap for more steps...

Step 4.3.1

Rewrite $0$ as $0^{8}$ .

$x = \pm \sqrt[8]{0^{8}}$

Step 4.3.2

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

$x = \pm 0$

Step 4.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4.4

Find the domain of $\frac{125}{x^{8}}$ .

Tap for more steps...

Step 4.4.1

Set the denominator in $\frac{125}{x^{8}}$ equal to $0$ to find where the expression is undefined.

$x^{8} = 0$

Step 4.4.2

Solve for $x$ .

Tap for more steps...

Step 4.4.2.1

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

$x = \pm \sqrt[8]{0}$

Step 4.4.2.2

Simplify $\pm \sqrt[8]{0}$ .

Tap for more steps...

Step 4.4.2.2.1

Rewrite $0$ as $0^{8}$ .

$x = \pm \sqrt[8]{0^{8}}$

Step 4.4.2.2.2

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

$x = \pm 0$

Step 4.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4.4.3

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

$(- \infty, 0) \cup (0, \infty)$

Step 4.5

Use each root to create test intervals.

$x < 0$

$x > 0$

Step 4.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 4.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

Tap for more steps...

Step 4.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 4.6.1.2

Replace $x$ with $- 2$ in the original inequality.

$\frac{125}{{(- 2)}^{8}} < 0$

Step 4.6.1.3

The left side $0.48828125$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 4.6.2

Test a value on the interval $x > 0$ to see if it makes the inequality true.

Tap for more steps...

Step 4.6.2.1

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

Step 4.6.2.2

Replace $x$ with $2$ in the original inequality.

$\frac{125}{{(2)}^{8}} < 0$

Step 4.6.2.3

The left side $0.48828125$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 4.6.3

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$x > 0$ False

$x < 0$ False

$x > 0$ False

Step 4.7

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 0$

Step 6