Linear Algebra Examples

$\frac{\sqrt[3]{5^{\frac{3}{2}}}}{\sqrt[3]{3^{\frac{3}{2}}}} = x$

Step 1

Rewrite the equation as $x = \frac{\sqrt[3]{5^{\frac{3}{2}}}}{\sqrt[3]{3^{\frac{3}{2}}}}$ .

$x = \frac{\sqrt[3]{5^{\frac{3}{2}}}}{\sqrt[3]{3^{\frac{3}{2}}}}$

Step 2

Simplify $\frac{\sqrt[3]{5^{\frac{3}{2}}}}{\sqrt[3]{3^{\frac{3}{2}}}}$ .

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite $5^{\frac{3}{2}}$ as ${(5^{\frac{1}{2}})}^{3}$ .

$x = \frac{\sqrt[3]{{(5^{\frac{1}{2}})}^{3}}}{\sqrt[3]{3^{\frac{3}{2}}}}$

Step 2.1.2

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

$x = \frac{5^{\frac{1}{2}}}{\sqrt[3]{3^{\frac{3}{2}}}}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Rewrite $3^{\frac{3}{2}}$ as ${(3^{\frac{1}{2}})}^{3}$ .

$x = \frac{5^{\frac{1}{2}}}{\sqrt[3]{{(3^{\frac{1}{2}})}^{3}}}$

Step 2.2.2

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

$x = \frac{5^{\frac{1}{2}}}{3^{\frac{1}{2}}}$

Step 3

The domain is the set of all valid $x$ values.

${x | x = \frac{5^{\frac{1}{2}}}{3^{\frac{1}{2}}}}$

Step 4