Finite Math Examples

$\frac{- 5 x}{\sqrt{x^{4}}} - \frac{4}{\sqrt{x}}$

Step 1

Set the denominator in $\frac{- 5 x}{\sqrt{x^{4}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x^{4}} = 0$

Step 2

Solve for $x$ .

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Step 2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{4}}}^{2} = 0^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4}}$ as $x^{\frac{4}{2}}$ .

${(x^{\frac{4}{2}})}^{2} = 0^{2}$

Step 2.2.2

Divide $4$ by $2$ .

${(x^{2})}^{2} = 0^{2}$

Step 2.2.3

Simplify the left side.

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Step 2.2.3.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.2.3.1.1

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

$x^{2 \cdot 2} = 0^{2}$

Step 2.2.3.1.2

Multiply $2$ by $2$ .

$x^{4} = 0^{2}$

Step 2.2.4

Simplify the right side.

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Step 2.2.4.1

Raising $0$ to any positive power yields $0$ .

$x^{4} = 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

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

$x = \pm \sqrt[4]{0}$

Step 2.3.2

Simplify $\pm \sqrt[4]{0}$ .

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Step 2.3.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 2.3.2.2

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

$x = \pm 0$

Step 2.3.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

Set the denominator in $\frac{4}{\sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x} = 0$

Step 4

Solve for $x$ .

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Step 4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

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

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$x = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 5

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

$x^{4} < 0$

Step 6

Solve for $x$ .

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Step 6.1

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

$\sqrt[4]{x^{4}} < \sqrt[4]{0}$

Step 6.2

Simplify the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Pull terms out from under the radical.

$| x | < \sqrt[4]{0}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\sqrt[4]{0}$ .

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Step 6.2.2.1.1

Rewrite $0$ as $0^{4}$ .

$| x | < \sqrt[4]{0^{4}}$

Step 6.2.2.1.2

Pull terms out from under the radical.

$| x | < | 0 |$

Step 6.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| x | < 0$

Step 6.3

Write $| x | < 0$ as a piecewise.

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Step 6.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 6.3.2

In the piece where $x$ is non-negative, remove the absolute value.

$x < 0$

Step 6.3.3

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x < 0$

Step 6.3.4

Write as a piecewise.

${\begin{cases} x < 0 & x \geq 0 \\ - x < 0 & x < 0 \end{cases}$

Step 6.4

Find the intersection of $x < 0$ and $x \geq 0$ .

No solution

Step 6.5

Solve $- x < 0$ when $x < 0$ .

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Step 6.5.1

Divide each term in $- x < 0$ by $- 1$ and simplify.

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Step 6.5.1.1

Divide each term in $- x < 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{0}{- 1}$

Step 6.5.1.2

Simplify the left side.

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Step 6.5.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{0}{- 1}$

Step 6.5.1.2.2

Divide $x$ by $1$ .

$x > \frac{0}{- 1}$

Step 6.5.1.3

Simplify the right side.

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Step 6.5.1.3.1

Divide $0$ by $- 1$ .

$x > 0$

Step 6.5.2

Find the intersection of $x > 0$ and $x < 0$ .

No solution

Step 6.6

Find the union of the solutions.

No solution

Step 7

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 8

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 $0$ .

$x \leq 0$

$(- \infty, 0]$

Step 9