Find the Product (5 square root of 2-4 square root of 3)(5 square root of 2-4 square root of 3)

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

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

Multiply $5$ by $5$ .

$25 \sqrt{2} \sqrt{2} + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

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

$25 ({\sqrt{2}}^{1} \sqrt{2}) + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

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

$25 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

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

$25 {\sqrt{2}}^{1 + 1} + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

Add $1$ and $1$ .

$25 {\sqrt{2}}^{2} + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

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

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

$25 {(2^{\frac{1}{2}})}^{2} + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

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

$25 \cdot 2^{\frac{1}{2} \cdot 2} + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

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

$25 \cdot 2^{\frac{2}{2}} + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$25 \cdot 2^{\frac{2}{2}} + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

Rewrite the expression.

$25 \cdot 2^{1} + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

Evaluate the exponent.

$25 \cdot 2 + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

Multiply $25$ by $2$ .

$50 + 5 \sqrt{2} (- 4 \sqrt{3}) - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

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

Multiply $- 4$ by $5$ .

$50 - 20 \sqrt{2} \sqrt{3} - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

Combine using the product rule for radicals.

$50 - 20 \sqrt{3 \cdot 2} - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

Multiply $3$ by $2$ .

$50 - 20 \sqrt{6} - 4 \sqrt{3} (5 \sqrt{2}) - 4 \sqrt{3} (- 4 \sqrt{3})$

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

Multiply $5$ by $- 4$ .

$50 - 20 \sqrt{6} - 20 \sqrt{3} \sqrt{2} - 4 \sqrt{3} (- 4 \sqrt{3})$

Combine using the product rule for radicals.

$50 - 20 \sqrt{6} - 20 \sqrt{2 \cdot 3} - 4 \sqrt{3} (- 4 \sqrt{3})$

Multiply $2$ by $3$ .

$50 - 20 \sqrt{6} - 20 \sqrt{6} - 4 \sqrt{3} (- 4 \sqrt{3})$

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

Multiply $- 4$ by $- 4$ .

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 \sqrt{3} \sqrt{3}$

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

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 ({\sqrt{3}}^{1} \sqrt{3})$

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

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 {\sqrt{3}}^{2}$

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

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

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 {(3^{\frac{1}{2}})}^{2}$

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

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 \cdot 3^{\frac{1}{2} \cdot 2}$

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

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 \cdot 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 \cdot 3^{\frac{2}{2}}$

Rewrite the expression.

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 \cdot 3^{1}$

Evaluate the exponent.

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 16 \cdot 3$

Multiply $16$ by $3$ .

$50 - 20 \sqrt{6} - 20 \sqrt{6} + 48$

Add $50$ and $48$ .

$98 - 20 \sqrt{6} - 20 \sqrt{6}$

Subtract $20 \sqrt{6}$ from $- 20 \sqrt{6}$ .

$98 - 40 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$98 - 40 \sqrt{6}$

Decimal Form:

$0.02041028 \dots$

Find the Product (5 square root of 2-4 square root of 3)(5 square root of 2-4 square root of 3)

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