Simplify the Radical Expression 4 square root of 15( square root of 6+ square root of 5)

$4 \sqrt{15} (\sqrt{6} + \sqrt{5})$

Step 1

Apply the distributive property.

$4 \sqrt{15} \sqrt{6} + 4 \sqrt{15} \sqrt{5}$

Step 2

Multiply $4 \sqrt{15} \sqrt{6}$ .

Combine using the product rule for radicals.

$4 \sqrt{6 \cdot 15} + 4 \sqrt{15} \sqrt{5}$

Multiply $6$ by $15$ .

$4 \sqrt{90} + 4 \sqrt{15} \sqrt{5}$

Step 3

Multiply $4 \sqrt{15} \sqrt{5}$ .

Combine using the product rule for radicals.

$4 \sqrt{90} + 4 \sqrt{5 \cdot 15}$

Multiply $5$ by $15$ .

$4 \sqrt{90} + 4 \sqrt{75}$

Step 4

Simplify each term.

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$4 \sqrt{9 (10)} + 4 \sqrt{75}$

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

$4 \sqrt{3^{2} \cdot 10} + 4 \sqrt{75}$

Pull terms out from under the radical.

$4 (| 3 | \sqrt{10}) + 4 \sqrt{75}$

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

$4 (3 \sqrt{10}) + 4 \sqrt{75}$

Multiply $3$ by $4$ .

$12 \sqrt{10} + 4 \sqrt{75}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$12 \sqrt{10} + 4 \sqrt{25 (3)}$

Rewrite $25$ as $5^{2}$ .

$12 \sqrt{10} + 4 \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$12 \sqrt{10} + 4 (| 5 | \sqrt{3})$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$12 \sqrt{10} + 4 (5 \sqrt{3})$

Multiply $5$ by $4$ .

$12 \sqrt{10} + 20 \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$12 \sqrt{10} + 20 \sqrt{3}$

Decimal Form:

$72.58834807 \dots$

Simplify the Radical Expression 4 square root of 15( square root of 6+ square root of 5)

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