Simplify the Radical Expression (2 square root of 8+3 square root of 5)(6 square root of 8+7 square root of 5)

$(2 \sqrt{8} + 3 \sqrt{5}) (6 \sqrt{8} + 7 \sqrt{5})$

Step 1

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$(2 \sqrt{4 (2)} + 3 \sqrt{5}) (6 \sqrt{8} + 7 \sqrt{5})$

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

$(2 \sqrt{2^{2} \cdot 2} + 3 \sqrt{5}) (6 \sqrt{8} + 7 \sqrt{5})$

Pull terms out from under the radical.

$(2 (| 2 | \sqrt{2}) + 3 \sqrt{5}) (6 \sqrt{8} + 7 \sqrt{5})$

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

$(2 (2 \sqrt{2}) + 3 \sqrt{5}) (6 \sqrt{8} + 7 \sqrt{5})$

Multiply $2$ by $2$ .

$(4 \sqrt{2} + 3 \sqrt{5}) (6 \sqrt{8} + 7 \sqrt{5})$

Step 2

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$(4 \sqrt{2} + 3 \sqrt{5}) (6 \sqrt{4 (2)} + 7 \sqrt{5})$

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

$(4 \sqrt{2} + 3 \sqrt{5}) (6 \sqrt{2^{2} \cdot 2} + 7 \sqrt{5})$

Pull terms out from under the radical.

$(4 \sqrt{2} + 3 \sqrt{5}) (6 (| 2 | \sqrt{2}) + 7 \sqrt{5})$

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

$(4 \sqrt{2} + 3 \sqrt{5}) (6 (2 \sqrt{2}) + 7 \sqrt{5})$

Multiply $2$ by $6$ .

$(4 \sqrt{2} + 3 \sqrt{5}) (12 \sqrt{2} + 7 \sqrt{5})$

Step 3

Expand $(4 \sqrt{2} + 3 \sqrt{5}) (12 \sqrt{2} + 7 \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

$4 \sqrt{2} (12 \sqrt{2} + 7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2} + 7 \sqrt{5})$

Apply the distributive property.

$4 \sqrt{2} (12 \sqrt{2}) + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2} + 7 \sqrt{5})$

Apply the distributive property.

$4 \sqrt{2} (12 \sqrt{2}) + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $4 \sqrt{2} (12 \sqrt{2})$ .

Multiply $12$ by $4$ .

$48 \sqrt{2} \sqrt{2} + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

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

$48 ({\sqrt{2}}^{1} \sqrt{2}) + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

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

$48 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

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

$48 {\sqrt{2}}^{1 + 1} + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Add $1$ and $1$ .

$48 {\sqrt{2}}^{2} + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

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

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

$48 {(2^{\frac{1}{2}})}^{2} + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

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

$48 \cdot 2^{\frac{1}{2} \cdot 2} + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

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

$48 \cdot 2^{\frac{2}{2}} + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Cancel the common factor of $2$ .

Cancel the common factor.

$48 \cdot 2^{\frac{2}{2}} + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Rewrite the expression.

$48 \cdot 2^{1} + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Evaluate the exponent.

$48 \cdot 2 + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Multiply $48$ by $2$ .

$96 + 4 \sqrt{2} (7 \sqrt{5}) + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Multiply $4 \sqrt{2} (7 \sqrt{5})$ .

Multiply $7$ by $4$ .

$96 + 28 \sqrt{2} \sqrt{5} + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Combine using the product rule for radicals.

$96 + 28 \sqrt{5 \cdot 2} + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Multiply $5$ by $2$ .

$96 + 28 \sqrt{10} + 3 \sqrt{5} (12 \sqrt{2}) + 3 \sqrt{5} (7 \sqrt{5})$

Multiply $3 \sqrt{5} (12 \sqrt{2})$ .

Multiply $12$ by $3$ .

$96 + 28 \sqrt{10} + 36 \sqrt{5} \sqrt{2} + 3 \sqrt{5} (7 \sqrt{5})$

Combine using the product rule for radicals.

$96 + 28 \sqrt{10} + 36 \sqrt{2 \cdot 5} + 3 \sqrt{5} (7 \sqrt{5})$

Multiply $2$ by $5$ .

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 3 \sqrt{5} (7 \sqrt{5})$

Multiply $3 \sqrt{5} (7 \sqrt{5})$ .

Multiply $7$ by $3$ .

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 \sqrt{5} \sqrt{5}$

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

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 ({\sqrt{5}}^{1} \sqrt{5})$

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

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

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

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 {\sqrt{5}}^{1 + 1}$

Add $1$ and $1$ .

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 {\sqrt{5}}^{2}$

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

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

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 {(5^{\frac{1}{2}})}^{2}$

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

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 \cdot 5^{\frac{1}{2} \cdot 2}$

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

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 \cdot 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 \cdot 5^{\frac{2}{2}}$

Rewrite the expression.

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 \cdot 5^{1}$

Evaluate the exponent.

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 21 \cdot 5$

Multiply $21$ by $5$ .

$96 + 28 \sqrt{10} + 36 \sqrt{10} + 105$

Add $96$ and $105$ .

$201 + 28 \sqrt{10} + 36 \sqrt{10}$

Add $28 \sqrt{10}$ and $36 \sqrt{10}$ .

$201 + 64 \sqrt{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$201 + 64 \sqrt{10}$

Decimal Form:

$403.38577025 \dots$

Simplify the Radical Expression (2 square root of 8+3 square root of 5)(6 square root of 8+7 square root of 5)

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