Algebra Examples

$\sqrt{3 x^{5}} \cdot \sqrt{15 x}$

Step 1

Rewrite $3 x^{5}$ as ${(x^{2})}^{2} \cdot (3 x)$ .

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Step 1.1

Factor out $x^{4}$ .

$\sqrt{3 (x^{4} x)} \cdot \sqrt{15 x}$

Step 1.2

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

$\sqrt{3 ({(x^{2})}^{2} x)} \cdot \sqrt{15 x}$

Step 1.3

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

$\sqrt{{(x^{2})}^{2} \cdot 3 x} \cdot \sqrt{15 x}$

Step 1.4

Add parentheses.

$\sqrt{{(x^{2})}^{2} \cdot (3 x)} \cdot \sqrt{15 x}$

Step 2

Pull terms out from under the radical.

$| x^{2} | \sqrt{3 x} \cdot \sqrt{15 x}$

Step 3

Remove non-negative terms from the absolute value.

$x^{2} \sqrt{3 x} \cdot \sqrt{15 x}$

Step 4

Multiply $x^{2} \sqrt{3 x} \sqrt{15 x}$ .

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Step 4.1

Combine using the product rule for radicals.

$x^{2} \sqrt{15 x (3 x)}$

Step 4.2

Multiply $3$ by $15$ .

$x^{2} \sqrt{45 x \cdot x}$

Step 4.3

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

$x^{2} \sqrt{45 (x^{1} x)}$

Step 4.4

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

$x^{2} \sqrt{45 (x^{1} x^{1})}$

Step 4.5

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

$x^{2} \sqrt{45 x^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$x^{2} \sqrt{45 x^{2}}$

Step 5

Rewrite $45 x^{2}$ as ${(3 x)}^{2} \cdot 5$ .

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Step 5.1

Factor $9$ out of $45$ .

$x^{2} \sqrt{9 (5) x^{2}}$

Step 5.2

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

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

Step 5.3

Move $5$ .

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

Step 5.4

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

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

Step 6

Pull terms out from under the radical.

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

Step 7

Rewrite using the commutative property of multiplication.

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

Step 8

Remove non-negative terms from the absolute value.

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