Linear Algebra Examples

$4 x \sqrt{2 x \sqrt[3]{3 x}}$

Step 1

Set the radicand in $\sqrt{2 x \sqrt[3]{3 x}}$ greater than or equal to $0$ to find where the expression is defined.

$2 x \sqrt[3]{3 x} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

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

${(2 x \sqrt[3]{3 x})}^{3} \geq 0^{3}$

Step 2.2

Simplify each side of the inequality.

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Step 2.2.1

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

${(2 x {(3 x)}^{\frac{1}{3}})}^{3} \geq 0^{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${(2 x {(3 x)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.2.1.1

Apply the product rule to $3 x$ .

${(2 x (3^{\frac{1}{3}} x^{\frac{1}{3}}))}^{3} \geq 0^{3}$

Step 2.2.2.1.2

Rewrite using the commutative property of multiplication.

${(2 \cdot 3^{\frac{1}{3}} x \cdot x^{\frac{1}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.3

Multiply $x$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 2.2.2.1.3.1

Move $x^{\frac{1}{3}}$ .

${(2 \cdot 3^{\frac{1}{3}} (x^{\frac{1}{3}} x))}^{3} \geq 0^{3}$

Step 2.2.2.1.3.2

Multiply $x^{\frac{1}{3}}$ by $x$ .

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Step 2.2.2.1.3.2.1

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

${(2 \cdot 3^{\frac{1}{3}} (x^{\frac{1}{3}} x^{1}))}^{3} \geq 0^{3}$

Step 2.2.2.1.3.2.2

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

${(2 \cdot 3^{\frac{1}{3}} x^{\frac{1}{3} + 1})}^{3} \geq 0^{3}$

Step 2.2.2.1.3.3

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

${(2 \cdot 3^{\frac{1}{3}} x^{\frac{1}{3} + \frac{3}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.3.4

Combine the numerators over the common denominator.

${(2 \cdot 3^{\frac{1}{3}} x^{\frac{1 + 3}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.3.5

Add $1$ and $3$ .

${(2 \cdot 3^{\frac{1}{3}} x^{\frac{4}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.2.1.4.1

Apply the product rule to $2 \cdot 3^{\frac{1}{3}} x^{\frac{4}{3}}$ .

${(2 \cdot 3^{\frac{1}{3}})}^{3} {(x^{\frac{4}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.4.2

Apply the product rule to $2 \cdot 3^{\frac{1}{3}}$ .

$2^{3} \cdot {(3^{\frac{1}{3}})}^{3} {(x^{\frac{4}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.5

Raise $2$ to the power of $3$ .

$8 \cdot {(3^{\frac{1}{3}})}^{3} {(x^{\frac{4}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.6

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

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Step 2.2.2.1.6.1

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

$8 \cdot 3^{\frac{1}{3} \cdot 3} {(x^{\frac{4}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.6.2

Cancel the common factor of $3$ .

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Step 2.2.2.1.6.2.1

Cancel the common factor.

$8 \cdot 3^{\frac{1}{3} \cdot 3} {(x^{\frac{4}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.6.2.2

Rewrite the expression.

$8 \cdot 3^{1} {(x^{\frac{4}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.7

Evaluate the exponent.

$8 \cdot 3 {(x^{\frac{4}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.8

Multiply $8$ by $3$ .

$24 {(x^{\frac{4}{3}})}^{3} \geq 0^{3}$

Step 2.2.2.1.9

Multiply the exponents in ${(x^{\frac{4}{3}})}^{3}$ .

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Step 2.2.2.1.9.1

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

$24 x^{\frac{4}{3} \cdot 3} \geq 0^{3}$

Step 2.2.2.1.9.2

Cancel the common factor of $3$ .

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Step 2.2.2.1.9.2.1

Cancel the common factor.

$24 x^{\frac{4}{3} \cdot 3} \geq 0^{3}$

Step 2.2.2.1.9.2.2

Rewrite the expression.

$24 x^{4} \geq 0^{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

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

$24 x^{4} \geq 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Divide each term in $24 x^{4} \geq 0$ by $24$ and simplify.

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Step 2.3.1.1

Divide each term in $24 x^{4} \geq 0$ by $24$ .

$\frac{24 x^{4}}{24} \geq \frac{0}{24}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $24$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{24 x^{4}}{24} \geq \frac{0}{24}$

Step 2.3.1.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} \geq \frac{0}{24}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $24$ .

$x^{4} \geq 0$

Step 2.3.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4