Simplify the Radical Expression ( square root of 3+5 square root of 2)(4 square root of 2-2 square root of 3)

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

Step 1

Expand $(\sqrt{3} + 5 \sqrt{2}) (4 \sqrt{2} - 2 \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{3} (4 \sqrt{2})$ .

Combine using the product rule for radicals.

$4 \sqrt{3 \cdot 2} + \sqrt{3} (- 2 \sqrt{3}) + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Multiply $3$ by $2$ .

$4 \sqrt{6} + \sqrt{3} (- 2 \sqrt{3}) + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

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

Raise $\sqrt{3}$ to the power of $1$ .

$4 \sqrt{6} - 2 ({\sqrt{3}}^{1} \sqrt{3}) + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$4 \sqrt{6} - 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

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

$4 \sqrt{6} - 2 {\sqrt{3}}^{1 + 1} + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Add $1$ and $1$ .

$4 \sqrt{6} - 2 {\sqrt{3}}^{2} + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$4 \sqrt{6} - 2 {(3^{\frac{1}{2}})}^{2} + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

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

$4 \sqrt{6} - 2 \cdot 3^{\frac{1}{2} \cdot 2} + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Combine $\frac{1}{2}$ and $2$ .

$4 \sqrt{6} - 2 \cdot 3^{\frac{2}{2}} + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 \sqrt{6} - 2 \cdot 3^{\frac{2}{2}} + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Rewrite the expression.

$4 \sqrt{6} - 2 \cdot 3^{1} + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Evaluate the exponent.

$4 \sqrt{6} - 2 \cdot 3 + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Multiply $- 2$ by $3$ .

$4 \sqrt{6} - 6 + 5 \sqrt{2} (4 \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Multiply $5 \sqrt{2} (4 \sqrt{2})$ .

Multiply $4$ by $5$ .

$4 \sqrt{6} - 6 + 20 \sqrt{2} \sqrt{2} + 5 \sqrt{2} (- 2 \sqrt{3})$

Raise $\sqrt{2}$ to the power of $1$ .

$4 \sqrt{6} - 6 + 20 ({\sqrt{2}}^{1} \sqrt{2}) + 5 \sqrt{2} (- 2 \sqrt{3})$

Raise $\sqrt{2}$ to the power of $1$ .

$4 \sqrt{6} - 6 + 20 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 5 \sqrt{2} (- 2 \sqrt{3})$

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

$4 \sqrt{6} - 6 + 20 {\sqrt{2}}^{1 + 1} + 5 \sqrt{2} (- 2 \sqrt{3})$

Add $1$ and $1$ .

$4 \sqrt{6} - 6 + 20 {\sqrt{2}}^{2} + 5 \sqrt{2} (- 2 \sqrt{3})$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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

$4 \sqrt{6} - 6 + 20 {(2^{\frac{1}{2}})}^{2} + 5 \sqrt{2} (- 2 \sqrt{3})$

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

$4 \sqrt{6} - 6 + 20 \cdot 2^{\frac{1}{2} \cdot 2} + 5 \sqrt{2} (- 2 \sqrt{3})$

Combine $\frac{1}{2}$ and $2$ .

$4 \sqrt{6} - 6 + 20 \cdot 2^{\frac{2}{2}} + 5 \sqrt{2} (- 2 \sqrt{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 \sqrt{6} - 6 + 20 \cdot 2^{\frac{2}{2}} + 5 \sqrt{2} (- 2 \sqrt{3})$

Rewrite the expression.

$4 \sqrt{6} - 6 + 20 \cdot 2^{1} + 5 \sqrt{2} (- 2 \sqrt{3})$

Evaluate the exponent.

$4 \sqrt{6} - 6 + 20 \cdot 2 + 5 \sqrt{2} (- 2 \sqrt{3})$

Multiply $20$ by $2$ .

$4 \sqrt{6} - 6 + 40 + 5 \sqrt{2} (- 2 \sqrt{3})$

Multiply $5 \sqrt{2} (- 2 \sqrt{3})$ .

Multiply $- 2$ by $5$ .

$4 \sqrt{6} - 6 + 40 - 10 \sqrt{2} \sqrt{3}$

Combine using the product rule for radicals.

$4 \sqrt{6} - 6 + 40 - 10 \sqrt{3 \cdot 2}$

Multiply $3$ by $2$ .

$4 \sqrt{6} - 6 + 40 - 10 \sqrt{6}$

Subtract $10 \sqrt{6}$ from $4 \sqrt{6}$ .

$- 6 + 40 - 6 \sqrt{6}$

Add $- 6$ and $40$ .

$34 - 6 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$34 - 6 \sqrt{6}$

Decimal Form:

$19.30306154 \dots$

Simplify the Radical Expression ( square root of 3+5 square root of 2)(4 square root of 2-2 square root of 3)

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