Simplify (4+2 square root of 5)(3 square root of 3+4 square root of 5)

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify each term.

Multiply $3$ by $4$ .

$12 \sqrt{3} + 4 (4 \sqrt{5}) + 2 \sqrt{5} (3 \sqrt{3}) + 2 \sqrt{5} (4 \sqrt{5})$

Multiply $4$ by $4$ .

$12 \sqrt{3} + 16 \sqrt{5} + 2 \sqrt{5} (3 \sqrt{3}) + 2 \sqrt{5} (4 \sqrt{5})$

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

Multiply $3$ by $2$ .

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{5} \sqrt{3} + 2 \sqrt{5} (4 \sqrt{5})$

Combine using the product rule for radicals.

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{3 \cdot 5} + 2 \sqrt{5} (4 \sqrt{5})$

Multiply $3$ by $5$ .

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 2 \sqrt{5} (4 \sqrt{5})$

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

Multiply $4$ by $2$ .

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 \sqrt{5} \sqrt{5}$

Raise $\sqrt{5}$ to the power of $1$ .

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 ({\sqrt{5}}^{1} \sqrt{5})$

Raise $\sqrt{5}$ to the power of $1$ .

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

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

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 {\sqrt{5}}^{1 + 1}$

Add $1$ and $1$ .

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 {\sqrt{5}}^{2}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 {(5^{\frac{1}{2}})}^{2}$

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

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 \cdot 5^{\frac{1}{2} \cdot 2}$

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

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 \cdot 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 \cdot 5^{\frac{2}{2}}$

Rewrite the expression.

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 \cdot 5^{1}$

Evaluate the exponent.

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 8 \cdot 5$

Multiply $8$ by $5$ .

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 40$

Step 3

The result can be shown in multiple forms.

Exact Form:

$12 \sqrt{3} + 16 \sqrt{5} + 6 \sqrt{15} + 40$

Decimal Form:

$119.79959740 \dots$

Simplify (4+2 square root of 5)(3 square root of 3+4 square root of 5)

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