Simplify the Radical Expression square root of 2a* square root of 14a^3* square root of 5a

$\sqrt{2 a} \cdot \sqrt{14 a^{3}} \cdot \sqrt{5 a}$

Step 1

Rewrite $14 a^{3}$ as $a^{2} \cdot (14 a)$ .

Factor out $a^{2}$ .

$\sqrt{2 a} \cdot \sqrt{14 (a^{2} a)} \cdot \sqrt{5 a}$

Reorder $14$ and $a^{2}$ .

$\sqrt{2 a} \cdot \sqrt{a^{2} \cdot 14 a} \cdot \sqrt{5 a}$

Add parentheses.

$\sqrt{2 a} \cdot \sqrt{a^{2} \cdot (14 a)} \cdot \sqrt{5 a}$

Step 2

Pull terms out from under the radical.

$\sqrt{2 a} \cdot (| a | \sqrt{14 a}) \cdot \sqrt{5 a}$

Step 3

Multiply $\sqrt{2 a} (| a | \sqrt{14 a})$ .

Combine using the product rule for radicals.

$\sqrt{14 a (2 a)} | a | \cdot \sqrt{5 a}$

Multiply $2$ by $14$ .

$\sqrt{28 a \cdot a} | a | \cdot \sqrt{5 a}$

Raise $a$ to the power of $1$ .

$\sqrt{28 (a^{1} a)} | a | \cdot \sqrt{5 a}$

Raise $a$ to the power of $1$ .

$\sqrt{28 (a^{1} a^{1})} | a | \cdot \sqrt{5 a}$

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

$\sqrt{28 a^{1 + 1}} | a | \cdot \sqrt{5 a}$

Add $1$ and $1$ .

$\sqrt{28 a^{2}} | a | \cdot \sqrt{5 a}$

Step 4

Rewrite $28 a^{2}$ as ${(2 a)}^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$\sqrt{4 (7) a^{2}} | a | \cdot \sqrt{5 a}$

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

$\sqrt{2^{2} \cdot 7 a^{2}} | a | \cdot \sqrt{5 a}$

Move $7$ .

$\sqrt{2^{2} a^{2} \cdot 7} | a | \cdot \sqrt{5 a}$

Rewrite $2^{2} a^{2}$ as ${(2 a)}^{2}$ .

$\sqrt{{(2 a)}^{2} \cdot 7} | a | \cdot \sqrt{5 a}$

Step 5

Pull terms out from under the radical.

$| 2 a | \sqrt{7} | a | \cdot \sqrt{5 a}$

Step 6

Remove non-negative terms from the absolute value.

$2 | a | \sqrt{7} | a | \cdot \sqrt{5 a}$

Step 7

Multiply $2 | a | \sqrt{7} | a |$ .

To multiply absolute values, multiply the terms inside each absolute value.

$2 | a \cdot a | \sqrt{7} \cdot \sqrt{5 a}$

Raise $a$ to the power of $1$ .

$2 | a^{1} a | \sqrt{7} \cdot \sqrt{5 a}$

Raise $a$ to the power of $1$ .

$2 | a^{1} a^{1} | \sqrt{7} \cdot \sqrt{5 a}$

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

$2 | a^{1 + 1} | \sqrt{7} \cdot \sqrt{5 a}$

Add $1$ and $1$ .

$2 | a^{2} | \sqrt{7} \cdot \sqrt{5 a}$

Step 8

Remove non-negative terms from the absolute value.

$2 a^{2} \sqrt{7} \cdot \sqrt{5 a}$

Step 9

Multiply $2 a^{2} \sqrt{7} \sqrt{5 a}$ .

Combine using the product rule for radicals.

$2 a^{2} \sqrt{5 a \cdot 7}$

Multiply $7$ by $5$ .

$2 a^{2} \sqrt{35 a}$

Simplify the Radical Expression square root of 2a* square root of 14a^3* square root of 5a

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