Simplify 5/( cube root of 7- cube root of 2)

$\frac{5}{\sqrt[3]{7} - \sqrt[3]{2}}$

Step 1

Multiply $\frac{5}{\sqrt[3]{7} - \sqrt[3]{2}}$ by $\frac{{\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2}}{{\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2}}$ .

$\frac{5}{\sqrt[3]{7} - \sqrt[3]{2}} \cdot \frac{{\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2}}{{\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2}}$

Step 2

Simplify terms.

$\frac{5 ({\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2})}{(\sqrt[3]{7} - \sqrt[3]{2}) ({\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2})}$

Expand the denominator using the FOIL method.

$\frac{5 ({\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2})}{{\sqrt[3]{7}}^{3} + \sqrt[3]{98} + \sqrt[3]{7} {(- \sqrt[3]{2})}^{2} - \sqrt[3]{2} {\sqrt[3]{7}}^{2} - \sqrt[3]{28} - \sqrt[3]{2} {(- \sqrt[3]{2})}^{2}}$

Simplify.

$\frac{5 ({\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2})}{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 ({\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2})}{5}$

Divide ${\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2}$ by $1$ .

${\sqrt[3]{7}}^{2} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2}$

Step 3

Simplify each term.

Rewrite ${\sqrt[3]{7}}^{2}$ as ${(7^{2})}^{\frac{1}{3}}$ .

$\sqrt[3]{7^{2}} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2}$

Raise $7$ to the power of $2$ .

$\sqrt[3]{49} - \sqrt[3]{7} (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2}$

Multiply $- \sqrt[3]{7} (- \sqrt[3]{2})$ .

Multiply $- 1$ by $- 1$ .

$\sqrt[3]{49} + 1 \sqrt[3]{7} \sqrt[3]{2} + {(- \sqrt[3]{2})}^{2}$

Multiply $\sqrt[3]{7}$ by $1$ .

$\sqrt[3]{49} + \sqrt[3]{7} \sqrt[3]{2} + {(- \sqrt[3]{2})}^{2}$

Combine using the product rule for radicals.

$\sqrt[3]{49} + \sqrt[3]{7 \cdot 2} + {(- \sqrt[3]{2})}^{2}$

Multiply $7$ by $2$ .

$\sqrt[3]{49} + \sqrt[3]{14} + {(- \sqrt[3]{2})}^{2}$

Apply the product rule to $- \sqrt[3]{2}$ .

$\sqrt[3]{49} + \sqrt[3]{14} + {(- 1)}^{2} {\sqrt[3]{2}}^{2}$

Raise $- 1$ to the power of $2$ .

$\sqrt[3]{49} + \sqrt[3]{14} + 1 {\sqrt[3]{2}}^{2}$

Multiply ${\sqrt[3]{2}}^{2}$ by $1$ .

$\sqrt[3]{49} + \sqrt[3]{14} + {\sqrt[3]{2}}^{2}$

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

$\sqrt[3]{49} + \sqrt[3]{14} + \sqrt[3]{2^{2}}$

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

$\sqrt[3]{49} + \sqrt[3]{14} + \sqrt[3]{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt[3]{49} + \sqrt[3]{14} + \sqrt[3]{4}$

Decimal Form:

$7.65684902 \dots$

Simplify 5/( cube root of 7- cube root of 2)

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