Algebra Examples

$\sqrt[3]{{(27 y^{3})}^{4}} \sqrt[6]{{(27 y^{3})}^{4}}$

Step 1

Simplify the expression.

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Step 1.1

Apply the product rule to $27 y^{3}$ .

$\sqrt[3]{27^{4} {(y^{3})}^{4}} \sqrt[6]{{(27 y^{3})}^{4}}$

Step 1.2

Raise $27$ to the power of $4$ .

$\sqrt[3]{531441 {(y^{3})}^{4}} \sqrt[6]{{(27 y^{3})}^{4}}$

Step 1.3

Multiply the exponents in ${(y^{3})}^{4}$ .

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Step 1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[3]{531441 y^{3 \cdot 4}} \sqrt[6]{{(27 y^{3})}^{4}}$

Step 1.3.2

Multiply $3$ by $4$ .

$\sqrt[3]{531441 y^{12}} \sqrt[6]{{(27 y^{3})}^{4}}$

Step 2

Rewrite $531441 y^{12}$ as ${(81 y^{4})}^{3}$ .

$\sqrt[3]{{(81 y^{4})}^{3}} \sqrt[6]{{(27 y^{3})}^{4}}$

Step 3

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

$81 y^{4} \sqrt[6]{{(27 y^{3})}^{4}}$

Step 4

Simplify the expression.

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Step 4.1

Apply the product rule to $27 y^{3}$ .

$81 y^{4} \sqrt[6]{27^{4} {(y^{3})}^{4}}$

Step 4.2

Raise $27$ to the power of $4$ .

$81 y^{4} \sqrt[6]{531441 {(y^{3})}^{4}}$

Step 4.3

Multiply the exponents in ${(y^{3})}^{4}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$81 y^{4} \sqrt[6]{531441 y^{3 \cdot 4}}$

Step 4.3.2

Multiply $3$ by $4$ .

$81 y^{4} \sqrt[6]{531441 y^{12}}$

Step 5

Rewrite $531441 y^{12}$ as ${(9 y^{2})}^{6}$ .

$81 y^{4} \sqrt[6]{{(9 y^{2})}^{6}}$

Step 6

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

$81 y^{4} (9 y^{2})$

Step 7

Rewrite using the commutative property of multiplication.

$81 \cdot 9 y^{4} y^{2}$

Step 8

Multiply $y^{4}$ by $y^{2}$ by adding the exponents.

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Step 8.1

Move $y^{2}$ .

$81 \cdot 9 (y^{2} y^{4})$

Step 8.2

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

$81 \cdot 9 y^{2 + 4}$

Step 8.3

Add $2$ and $4$ .

$81 \cdot 9 y^{6}$

Step 9

Multiply $81$ by $9$ .

$729 y^{6}$