Simplify the Radical Expression 2 square root of 20+ cube root of 16- square root of 45+ cube root of 54

$2 \sqrt{20} + \sqrt[3]{16} - \sqrt{45} + \sqrt[3]{54}$

Step 1

Simplify each term.

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$2 \sqrt{4 (5)} + \sqrt[3]{16} - \sqrt{45} + \sqrt[3]{54}$

Rewrite $4$ as $2^{2}$ .

$2 \sqrt{2^{2} \cdot 5} + \sqrt[3]{16} - \sqrt{45} + \sqrt[3]{54}$

Pull terms out from under the radical.

$2 (| 2 | \sqrt{5}) + \sqrt[3]{16} - \sqrt{45} + \sqrt[3]{54}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$2 (2 \sqrt{5}) + \sqrt[3]{16} - \sqrt{45} + \sqrt[3]{54}$

Multiply $2$ by $2$ .

$4 \sqrt{5} + \sqrt[3]{16} - \sqrt{45} + \sqrt[3]{54}$

Rewrite $16$ as $2^{3} \cdot 2$ .

Factor $8$ out of $16$ .

$4 \sqrt{5} + \sqrt[3]{8 (2)} - \sqrt{45} + \sqrt[3]{54}$

Rewrite $8$ as $2^{3}$ .

$4 \sqrt{5} + \sqrt[3]{2^{3} \cdot 2} - \sqrt{45} + \sqrt[3]{54}$

Pull terms out from under the radical.

$4 \sqrt{5} + 2 \sqrt[3]{2} - \sqrt{45} + \sqrt[3]{54}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$4 \sqrt{5} + 2 \sqrt[3]{2} - \sqrt{9 (5)} + \sqrt[3]{54}$

Rewrite $9$ as $3^{2}$ .

$4 \sqrt{5} + 2 \sqrt[3]{2} - \sqrt{3^{2} \cdot 5} + \sqrt[3]{54}$

Pull terms out from under the radical.

$4 \sqrt{5} + 2 \sqrt[3]{2} - (| 3 | \sqrt{5}) + \sqrt[3]{54}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$4 \sqrt{5} + 2 \sqrt[3]{2} - (3 \sqrt{5}) + \sqrt[3]{54}$

Multiply $3$ by $- 1$ .

$4 \sqrt{5} + 2 \sqrt[3]{2} - 3 \sqrt{5} + \sqrt[3]{54}$

Rewrite $54$ as $3^{3} \cdot 2$ .

Factor $27$ out of $54$ .

$4 \sqrt{5} + 2 \sqrt[3]{2} - 3 \sqrt{5} + \sqrt[3]{27 (2)}$

Rewrite $27$ as $3^{3}$ .

$4 \sqrt{5} + 2 \sqrt[3]{2} - 3 \sqrt{5} + \sqrt[3]{3^{3} \cdot 2}$

Pull terms out from under the radical.

$4 \sqrt{5} + 2 \sqrt[3]{2} - 3 \sqrt{5} + 3 \sqrt[3]{2}$

Step 2

Subtract $3 \sqrt{5}$ from $4 \sqrt{5}$ .

$\sqrt{5} + 2 \sqrt[3]{2} + 3 \sqrt[3]{2}$

Step 3

Add $2 \sqrt[3]{2}$ and $3 \sqrt[3]{2}$ .

$\sqrt{5} + 5 \sqrt[3]{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{5} + 5 \sqrt[3]{2}$

Decimal Form:

$8.53567322 \dots$

Simplify the Radical Expression 2 square root of 20+ cube root of 16- square root of 45+ cube root of 54

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