Finite Math Examples

$\frac{\log (\sqrt{x \sqrt{x}})}{\log (\sqrt[3]{x})}$

Step 1

Set the denominator in $\frac{\log (\sqrt{x \sqrt{x}})}{\log (\sqrt[3]{x})}$ equal to $0$ to find where the expression is undefined.

$\log (\sqrt[3]{x}) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite $\log (\sqrt[3]{x}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{0} = \sqrt[3]{x}$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Rewrite the equation as $\sqrt[3]{x} = 10^{0}$ .

$\sqrt[3]{x} = 10^{0}$

Step 2.2.2

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

${\sqrt[3]{x}}^{3} = {(10^{0})}^{3}$

Step 2.2.3

Simplify each side of the equation.

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Step 2.2.3.1

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

${(x^{\frac{1}{3}})}^{3} = {(10^{0})}^{3}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

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

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Step 2.2.3.2.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 2.2.3.2.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = {(10^{0})}^{3}$

Step 2.2.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.3.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(10^{0})}^{3}$

Step 2.2.3.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(10^{0})}^{3}$

Step 2.2.3.2.1.2

Simplify.

$x = {(10^{0})}^{3}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Simplify ${(10^{0})}^{3}$ .

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Step 2.2.3.3.1.1

Multiply the exponents in ${(10^{0})}^{3}$ .

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Step 2.2.3.3.1.1.1

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

$x = 10^{0 \cdot 3}$

Step 2.2.3.3.1.1.2

Multiply $0$ by $3$ .

$x = 10^{0}$

Step 2.2.3.3.1.2

Anything raised to $0$ is $1$ .

$x = 1$

Step 3

Set the argument in $\log (\sqrt{x \sqrt{x}})$ less than or equal to $0$ to find where the expression is undefined.

$\sqrt{x \sqrt{x}} \leq 0$

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

Simplify each side of the inequality.

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

${(x \sqrt{x})}^{\frac{1}{2} \cdot 2} \leq 0^{2}$

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.

${(x \sqrt{x})}^{\frac{1}{2} \cdot 2} \leq 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(x \sqrt{x})}^{1} \leq 0^{2}$

Step 4.2.2.1.2

Simplify.

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$x \sqrt{x} \leq 0$

Step 4.3

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

${(x \sqrt{x})}^{2} \leq 0^{2}$

Step 4.4

Simplify each side of the inequality.

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Step 4.4.1

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

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

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

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

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Step 4.4.2.1.1

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.4.2.1.1.1

Multiply $x$ by $x^{\frac{1}{2}}$ .

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Step 4.4.2.1.1.1.1

Raise $x$ to the power of $1$ .

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

Step 4.4.2.1.1.1.2

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

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

Step 4.4.2.1.1.2

Write $1$ as a fraction with a common denominator.

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

Step 4.4.2.1.1.3

Combine the numerators over the common denominator.

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

Step 4.4.2.1.1.4

Add $2$ and $1$ .

${(x^{\frac{3}{2}})}^{2} \leq 0^{2}$

Step 4.4.2.1.2

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

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Step 4.4.2.1.2.1

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

$x^{\frac{3}{2} \cdot 2} \leq 0^{2}$

Step 4.4.2.1.2.2

Cancel the common factor of $2$ .

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Step 4.4.2.1.2.2.1

Cancel the common factor.

$x^{\frac{3}{2} \cdot 2} \leq 0^{2}$

Step 4.4.2.1.2.2.2

Rewrite the expression.

$x^{3} \leq 0^{2}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

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

$x^{3} \leq 0$

Step 4.5

Solve for $x$ .

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Step 4.5.1

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

$\sqrt[3]{x^{3}} \leq \sqrt[3]{0}$

Step 4.5.2

Simplify the equation.

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Step 4.5.2.1

Simplify the left side.

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Step 4.5.2.1.1

Pull terms out from under the radical.

$x \leq \sqrt[3]{0}$

Step 4.5.2.2

Simplify the right side.

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Step 4.5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 4.5.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x \leq \sqrt[3]{0^{3}}$

Step 4.5.2.2.1.2

Pull terms out from under the radical.

$x \leq 0$

Step 5

Set the argument in $\log (\sqrt[3]{x})$ less than or equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x} \leq 0$

Step 6

Solve for $x$ .

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Step 6.1

To remove the radical on the left side of the inequality, cube both sides of the inequality.

${\sqrt[3]{x}}^{3} \leq 0^{3}$

Step 6.2

Simplify each side of the inequality.

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Step 6.2.1

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

${(x^{\frac{1}{3}})}^{3} \leq 0^{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 6.2.2.1.1.1

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

$x^{\frac{1}{3} \cdot 3} \leq 0^{3}$

Step 6.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} \leq 0^{3}$

Step 6.2.2.1.1.2.2

Rewrite the expression.

$x^{1} \leq 0^{3}$

Step 6.2.2.1.2

Simplify.

$x \leq 0^{3}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

$x \leq 0$

Step 7

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

$x < 0$

Step 8

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

$x \sqrt{x} < 0$

Step 9

Solve for $x$ .

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Step 9.1

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

${(x \sqrt{x})}^{2} < 0^{2}$

Step 9.2

Simplify each side of the inequality.

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Step 9.2.1

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

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

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

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

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Step 9.2.2.1.1

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 9.2.2.1.1.1

Multiply $x$ by $x^{\frac{1}{2}}$ .

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Step 9.2.2.1.1.1.1

Raise $x$ to the power of $1$ .

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

Step 9.2.2.1.1.1.2

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

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

Step 9.2.2.1.1.2

Write $1$ as a fraction with a common denominator.

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

Step 9.2.2.1.1.3

Combine the numerators over the common denominator.

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

Step 9.2.2.1.1.4

Add $2$ and $1$ .

${(x^{\frac{3}{2}})}^{2} < 0^{2}$

Step 9.2.2.1.2

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

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Step 9.2.2.1.2.1

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

$x^{\frac{3}{2} \cdot 2} < 0^{2}$

Step 9.2.2.1.2.2

Cancel the common factor of $2$ .

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Step 9.2.2.1.2.2.1

Cancel the common factor.

$x^{\frac{3}{2} \cdot 2} < 0^{2}$

Step 9.2.2.1.2.2.2

Rewrite the expression.

$x^{3} < 0^{2}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

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

$x^{3} < 0$

Step 9.3

Solve for $x$ .

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Step 9.3.1

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

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 9.3.2

Simplify the equation.

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Step 9.3.2.1

Simplify the left side.

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Step 9.3.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 9.3.2.2

Simplify the right side.

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Step 9.3.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 9.3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 9.3.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 9.4

Find the domain of $x \sqrt{x}$ .

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Step 9.4.1

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

$x \geq 0$

Step 9.4.2

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

$[0, \infty)$

Step 9.5

Use each root to create test intervals.

$x < 0$

$x > 0$

Step 9.6

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

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Step 9.6.1

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

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Step 9.6.1.1

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

$x = - 2$

Step 9.6.1.2

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

$(- 2) \sqrt{- 2} < 0$

Step 9.6.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 9.6.2

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

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Step 9.6.2.1

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

$x = 2$

Step 9.6.2.2

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

$(2) \sqrt{2} < 0$

Step 9.6.2.3

The left side $2.82842712$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 9.6.3

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

$x < 0$ False

$x > 0$ False

$x < 0$ False

$x > 0$ False

Step 9.7

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

No solution

Step 10

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 $0$ .

$x \leq 0, x = 1$

$(- \infty, 0] \cup [1, 1]$

Step 11