Algebra Examples

$\sqrt{24 b^{3}} \cdot \sqrt{40 b^{2}} \cdot \sqrt{b^{2}}$

Step 1

Rewrite $24 b^{3}$ as ${(2 b)}^{2} \cdot (6 b)$ .

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Step 1.1

Factor $4$ out of $24$ .

$\sqrt{4 (6) b^{3}} \cdot \sqrt{40 b^{2}} \cdot \sqrt{b^{2}}$

Step 1.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 6 b^{3}} \cdot \sqrt{40 b^{2}} \cdot \sqrt{b^{2}}$

Step 1.3

Factor out $b^{2}$ .

$\sqrt{2^{2} \cdot 6 (b^{2} b)} \cdot \sqrt{40 b^{2}} \cdot \sqrt{b^{2}}$

Step 1.4

Move $6$ .

$\sqrt{2^{2} b^{2} \cdot 6 b} \cdot \sqrt{40 b^{2}} \cdot \sqrt{b^{2}}$

Step 1.5

Rewrite $2^{2} b^{2}$ as ${(2 b)}^{2}$ .

$\sqrt{{(2 b)}^{2} \cdot 6 b} \cdot \sqrt{40 b^{2}} \cdot \sqrt{b^{2}}$

Step 1.6

Add parentheses.

$\sqrt{{(2 b)}^{2} \cdot (6 b)} \cdot \sqrt{40 b^{2}} \cdot \sqrt{b^{2}}$

Step 2

Pull terms out from under the radical.

$2 b \sqrt{6 b} \cdot \sqrt{40 b^{2}} \cdot \sqrt{b^{2}}$

Step 3

Rewrite $40 b^{2}$ as ${(2 b)}^{2} \cdot 10$ .

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Step 3.1

Factor $4$ out of $40$ .

$2 b \sqrt{6 b} \cdot \sqrt{4 (10) b^{2}} \cdot \sqrt{b^{2}}$

Step 3.2

Rewrite $4$ as $2^{2}$ .

$2 b \sqrt{6 b} \cdot \sqrt{2^{2} \cdot 10 b^{2}} \cdot \sqrt{b^{2}}$

Step 3.3

Move $10$ .

$2 b \sqrt{6 b} \cdot \sqrt{2^{2} b^{2} \cdot 10} \cdot \sqrt{b^{2}}$

Step 3.4

Rewrite $2^{2} b^{2}$ as ${(2 b)}^{2}$ .

$2 b \sqrt{6 b} \cdot \sqrt{{(2 b)}^{2} \cdot 10} \cdot \sqrt{b^{2}}$

Step 4

Pull terms out from under the radical.

$2 b \sqrt{6 b} \cdot (2 b \sqrt{10}) \cdot \sqrt{b^{2}}$

Step 5

Multiply $b$ by $b$ by adding the exponents.

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Step 5.1

Move $b$ .

$2 (b \cdot b) \sqrt{6 b} \cdot (2 \sqrt{10}) \cdot \sqrt{b^{2}}$

Step 5.2

Multiply $b$ by $b$ .

$2 b^{2} \sqrt{6 b} \cdot (2 \sqrt{10}) \cdot \sqrt{b^{2}}$

Step 6

Multiply $2 b^{2} \sqrt{6 b} (2 \sqrt{10})$ .

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Step 6.1

Multiply $2$ by $2$ .

$4 b^{2} \sqrt{6 b} \sqrt{10} \cdot \sqrt{b^{2}}$

Step 6.2

Combine using the product rule for radicals.

$4 b^{2} \sqrt{10 (6 b)} \cdot \sqrt{b^{2}}$

Step 6.3

Multiply $10$ by $6$ .

$4 b^{2} \sqrt{60 b} \cdot \sqrt{b^{2}}$

Step 7

Rewrite $60 b$ as $2^{2} \cdot (15 b)$ .

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Step 7.1

Factor $4$ out of $60$ .

$4 b^{2} \sqrt{4 (15) b} \cdot \sqrt{b^{2}}$

Step 7.2

Rewrite $4$ as $2^{2}$ .

$4 b^{2} \sqrt{2^{2} \cdot 15 b} \cdot \sqrt{b^{2}}$

Step 7.3

Add parentheses.

$4 b^{2} \sqrt{2^{2} \cdot (15 b)} \cdot \sqrt{b^{2}}$

Step 8

Pull terms out from under the radical.

$4 b^{2} (2 \sqrt{15 b}) \cdot \sqrt{b^{2}}$

Step 9

Multiply $2$ by $4$ .

$8 b^{2} \sqrt{15 b} \cdot \sqrt{b^{2}}$

Step 10

Pull terms out from under the radical, assuming positive real numbers.

$8 b^{2} \sqrt{15 b} \cdot b$

Step 11

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

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Step 11.1

Move $b$ .

$8 (b \cdot b^{2}) \sqrt{15 b}$

Step 11.2

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

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Step 11.2.1

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

$8 (b^{1} b^{2}) \sqrt{15 b}$

Step 11.2.2

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

$8 b^{1 + 2} \sqrt{15 b}$

Step 11.3

Add $1$ and $2$ .

$8 b^{3} \sqrt{15 b}$