Pre-Algebra Examples

$\sqrt{5 c d^{4}} \cdot \sqrt{14 c^{2} d^{3}}$

Step 1

Rewrite $5 c d^{4}$ as ${(d^{2})}^{2} \cdot (5 c)$ .

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Step 1.1

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

$\sqrt{5 c {(d^{2})}^{2}} \cdot \sqrt{14 c^{2} d^{3}}$

Step 1.2

Move $c$ .

$\sqrt{5 {(d^{2})}^{2} c} \cdot \sqrt{14 c^{2} d^{3}}$

Step 1.3

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

$\sqrt{{(d^{2})}^{2} \cdot 5 c} \cdot \sqrt{14 c^{2} d^{3}}$

Step 1.4

Add parentheses.

$\sqrt{{(d^{2})}^{2} \cdot (5 c)} \cdot \sqrt{14 c^{2} d^{3}}$

Step 2

Pull terms out from under the radical.

$d^{2} \sqrt{5 c} \cdot \sqrt{14 c^{2} d^{3}}$

Step 3

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

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Step 3.1

Factor out $d^{2}$ .

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

Step 3.2

Move $14$ .

$d^{2} \sqrt{5 c} \cdot \sqrt{c^{2} d^{2} \cdot 14 d}$

Step 3.3

Rewrite $c^{2} d^{2}$ as ${(c d)}^{2}$ .

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

Step 3.4

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $d$ .

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

Step 5.2

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

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Step 5.2.1

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

$d^{1} d^{2} \sqrt{5 c} \cdot (c \sqrt{14 d})$

Step 5.2.2

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

$d^{1 + 2} \sqrt{5 c} \cdot (c \sqrt{14 d})$

Step 5.3

Add $1$ and $2$ .

$d^{3} \sqrt{5 c} \cdot (c \sqrt{14 d})$

Step 6

Multiply $d^{3} \sqrt{5 c} (c \sqrt{14 d})$ .

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Step 6.1

Combine using the product rule for radicals.

$d^{3} (c \sqrt{5 c (14 d)})$

Step 6.2

Multiply $14$ by $5$ .

$d^{3} (c \sqrt{70 c d})$

Step 7

Reorder factors in $d^{3} (c \sqrt{70 c d})$ .

$d^{3} c \sqrt{70 c d}$