Algebra Examples

$\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{10 a^{2} b^{7}}$

Step 1

Rewrite $10 a^{2} b^{7}$ as $b^{4} \cdot (10 a^{2} b^{3})$ .

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Step 1.1

Factor out $b^{4}$ .

$\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{10 a^{2} (b^{4} b^{3})}$

Step 1.2

Move $a^{2}$ .

$\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{10 b^{4} a^{2} b^{3}}$

Step 1.3

Reorder $10$ and $b^{4}$ .

$\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{b^{4} \cdot 10 a^{2} b^{3}}$

Step 1.4

Add parentheses.

$\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{b^{4} \cdot 10 (a^{2} b^{3})}$

Step 1.5

Add parentheses.

$\sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{b^{4} \cdot (10 a^{2} b^{3})}$

Step 2

Pull terms out from under the radical.

$\sqrt[4]{8 a^{3} b} \cdot (b \sqrt[4]{10 a^{2} b^{3}})$

Step 3

Multiply $\sqrt[4]{8 a^{3} b} (b \sqrt[4]{10 a^{2} b^{3}})$ .

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Step 3.1

Combine using the product rule for radicals.

$b \sqrt[4]{8 a^{3} b (10 a^{2} b^{3})}$

Step 3.2

Multiply $10$ by $8$ .

$b \sqrt[4]{80 a^{3} b (a^{2} b^{3})}$

Step 3.3

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

$b \sqrt[4]{80 a^{2 + 3} b \cdot b^{3}}$

Step 3.4

Add $2$ and $3$ .

$b \sqrt[4]{80 a^{5} b \cdot b^{3}}$

Step 3.5

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

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Step 3.5.1

Move $b^{3}$ .

$b \sqrt[4]{80 a^{5} (b^{3} b)}$

Step 3.5.2

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

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Step 3.5.2.1

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

$b \sqrt[4]{80 a^{5} (b^{3} b^{1})}$

Step 3.5.2.2

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

$b \sqrt[4]{80 a^{5} b^{3 + 1}}$

Step 3.5.3

Add $3$ and $1$ .

$b \sqrt[4]{80 a^{5} b^{4}}$

Step 4

Rewrite $80 a^{5} b^{4}$ as ${(2 a b)}^{4} \cdot (5 a)$ .

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Step 4.1

Factor $16$ out of $80$ .

$b \sqrt[4]{16 (5) a^{5} b^{4}}$

Step 4.2

Rewrite $16$ as $2^{4}$ .

$b \sqrt[4]{2^{4} \cdot 5 a^{5} b^{4}}$

Step 4.3

Factor out $a^{4}$ .

$b \sqrt[4]{2^{4} \cdot 5 (a^{4} a) b^{4}}$

Step 4.4

Move $a$ .

$b \sqrt[4]{2^{4} \cdot 5 (a^{4}) b^{4} a}$

Step 4.5

Move $5$ .

$b \sqrt[4]{(2^{4} (a^{4})) b^{4} \cdot 5 a}$

Step 4.6

Rewrite $(2^{4} (a^{4})) b^{4}$ as ${(2 a b)}^{4}$ .

$b \sqrt[4]{{(2 a b)}^{4} \cdot 5 a}$

Step 4.7

Add parentheses.

$b \sqrt[4]{{(2 a b)}^{4} \cdot (5 a)}$

Step 5

Pull terms out from under the radical.

$b (2 a b \sqrt[4]{5 a})$

Step 6

Rewrite using the commutative property of multiplication.

$2 b (a b \sqrt[4]{5 a})$

Step 7

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

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Step 7.1

Move $b$ .

$2 (b \cdot b) (a \sqrt[4]{5 a})$

Step 7.2

Multiply $b$ by $b$ .

$2 b^{2} (a \sqrt[4]{5 a})$

$2 b^{2} a \sqrt[4]{5 a}$