Pre-Algebra Examples

$\sqrt{5 x^{5}} \sqrt{15 x^{2}}$

Step 1

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

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Step 1.1

Factor out $x^{4}$ .

$\sqrt{5 (x^{4} x)} \sqrt{15 x^{2}}$

Step 1.2

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

$\sqrt{5 ({(x^{2})}^{2} x)} \sqrt{15 x^{2}}$

Step 1.3

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

$\sqrt{{(x^{2})}^{2} \cdot 5 x} \sqrt{15 x^{2}}$

Step 1.4

Add parentheses.

$\sqrt{{(x^{2})}^{2} \cdot (5 x)} \sqrt{15 x^{2}}$

Step 2

Pull terms out from under the radical.

$x^{2} \sqrt{5 x} \sqrt{15 x^{2}}$

Step 3

Reorder $15$ and $x^{2}$ .

$x^{2} \sqrt{5 x} \sqrt{x^{2} \cdot 15}$

Step 4

Pull terms out from under the radical.

$x^{2} \sqrt{5 x} (x \sqrt{15})$

Step 5

Rewrite using the commutative property of multiplication.

$\sqrt{15} x^{2} \sqrt{5 x} x$

Step 6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 6.1

Move $x$ .

$\sqrt{15} (x \cdot x^{2}) \sqrt{5 x}$

Step 6.2

Multiply $x$ by $x^{2}$ .

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Step 6.2.1

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

$\sqrt{15} (x^{1} x^{2}) \sqrt{5 x}$

Step 6.2.2

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

$\sqrt{15} x^{1 + 2} \sqrt{5 x}$

Step 6.3

Add $1$ and $2$ .

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

Step 7

Multiply $\sqrt{15} x^{3} \sqrt{5 x}$ .

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Step 7.1

Combine using the product rule for radicals.

$x^{3} \sqrt{15 (5 x)}$

Step 7.2

Multiply $15$ by $5$ .

$x^{3} \sqrt{75 x}$

Step 8

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

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Step 8.1

Factor $25$ out of $75$ .

$x^{3} \sqrt{25 (3) x}$

Step 8.2

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

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

Step 8.3

Add parentheses.

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

Step 9

Pull terms out from under the radical.

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

Step 10

Rewrite using the commutative property of multiplication.

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