Pre-Algebra Examples

$\sqrt{5 x^{7}} \cdot \sqrt{35 x}$

Step 1

Rewrite $5 x^{7}$ as ${(x^{3})}^{2} \cdot (5 x)$ .

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Step 1.1

Factor out $x^{6}$ .

$\sqrt{5 (x^{6} x)} \cdot \sqrt{35 x}$

Step 1.2

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$\sqrt{5 ({(x^{3})}^{2} x)} \cdot \sqrt{35 x}$

Step 1.3

Reorder $5$ and ${(x^{3})}^{2}$ .

$\sqrt{{(x^{3})}^{2} \cdot 5 x} \cdot \sqrt{35 x}$

Step 1.4

Add parentheses.

$\sqrt{{(x^{3})}^{2} \cdot (5 x)} \cdot \sqrt{35 x}$

Step 2

Pull terms out from under the radical.

$x^{3} \sqrt{5 x} \cdot \sqrt{35 x}$

Step 3

Multiply $x^{3} \sqrt{5 x} \sqrt{35 x}$ .

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Step 3.1

Combine using the product rule for radicals.

$x^{3} \sqrt{35 x (5 x)}$

Step 3.2

Multiply $5$ by $35$ .

$x^{3} \sqrt{175 x \cdot x}$

Step 3.3

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

$x^{3} \sqrt{175 (x^{1} x)}$

Step 3.4

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

$x^{3} \sqrt{175 (x^{1} x^{1})}$

Step 3.5

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

$x^{3} \sqrt{175 x^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

$x^{3} \sqrt{175 x^{2}}$

Step 4

Rewrite $175 x^{2}$ as ${(5 x)}^{2} \cdot 7$ .

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Step 4.1

Factor $25$ out of $175$ .

$x^{3} \sqrt{25 (7) x^{2}}$

Step 4.2

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

$x^{3} \sqrt{5^{2} \cdot 7 x^{2}}$

Step 4.3

Move $7$ .

$x^{3} \sqrt{5^{2} x^{2} \cdot 7}$

Step 4.4

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

$x^{3} \sqrt{{(5 x)}^{2} \cdot 7}$

Step 5

Pull terms out from under the radical.

$x^{3} (5 x \sqrt{7})$

Step 6

Rewrite using the commutative property of multiplication.

$5 \sqrt{7} x^{3} x$

Step 7

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 7.1

Move $x$ .

$5 \sqrt{7} (x \cdot x^{3})$

Step 7.2

Multiply $x$ by $x^{3}$ .

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Step 7.2.1

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

$5 \sqrt{7} (x^{1} x^{3})$

Step 7.2.2

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

$5 \sqrt{7} x^{1 + 3}$

Step 7.3

Add $1$ and $3$ .

$5 \sqrt{7} x^{4}$