Algebra Examples

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

Step 1

Rewrite $3 x^{4}$ as ${(x^{2})}^{2} \cdot 3$ .

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Step 1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\sqrt{3 {(x^{2})}^{2}} \cdot \sqrt{5 x^{2}} \cdot \sqrt{10}$

Step 1.2

Reorder $3$ and ${(x^{2})}^{2}$ .

$\sqrt{{(x^{2})}^{2} \cdot 3} \cdot \sqrt{5 x^{2}} \cdot \sqrt{10}$

Step 2

Pull terms out from under the radical.

$| x^{2} | \sqrt{3} \cdot \sqrt{5 x^{2}} \cdot \sqrt{10}$

Step 3

Remove non-negative terms from the absolute value.

$x^{2} \sqrt{3} \cdot \sqrt{5 x^{2}} \cdot \sqrt{10}$

Step 4

Reorder $5$ and $x^{2}$ .

$x^{2} \sqrt{3} \cdot \sqrt{x^{2} \cdot 5} \cdot \sqrt{10}$

Step 5

Pull terms out from under the radical.

$x^{2} \sqrt{3} \cdot (| x | \sqrt{5}) \cdot \sqrt{10}$

Step 6

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

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Step 6.1

Combine using the product rule for radicals.

$x^{2} \sqrt{5 \cdot 3} | x | \cdot \sqrt{10}$

Step 6.2

Multiply $5$ by $3$ .

$x^{2} \sqrt{15} | x | \cdot \sqrt{10}$

Step 7

Multiply $x^{2} \sqrt{15} | x | \sqrt{10}$ .

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Step 7.1

Combine using the product rule for radicals.

$x^{2} \sqrt{10 \cdot 15} | x |$

Step 7.2

Multiply $10$ by $15$ .

$x^{2} \sqrt{150} | x |$

Step 8

Rewrite $150$ as $5^{2} \cdot 6$ .

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Step 8.1

Factor $25$ out of $150$ .

$x^{2} \sqrt{25 (6)} | x |$

Step 8.2

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

$x^{2} \sqrt{5^{2} \cdot 6} | x |$

Step 9

Pull terms out from under the radical.

$x^{2} (| 5 | \sqrt{6}) | x |$

Step 10

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

$x^{2} (5 \sqrt{6}) | x |$

Step 11

Move $5$ to the left of $x^{2}$ .

$5 x^{2} \sqrt{6} | x |$