Algebra Examples

$\sqrt[6]{64 {(2 y + 1)}^{18}}$

Step 1

Rewrite $64 {(2 y + 1)}^{18}$ as ${(2 {(2 y + 1)}^{3})}^{6}$ .

$\sqrt[6]{{(2 {(2 y + 1)}^{3})}^{6}}$

Step 2

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

$2 {(2 y + 1)}^{3}$

Step 3

Use the Binomial Theorem.

$2 ({(2 y)}^{3} + 3 {(2 y)}^{2} \cdot 1 + 3 (2 y) \cdot 1^{2} + 1^{3})$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the product rule to $2 y$ .

$2 (2^{3} y^{3} + 3 {(2 y)}^{2} \cdot 1 + 3 (2 y) \cdot 1^{2} + 1^{3})$

Step 4.1.2

Raise $2$ to the power of $3$ .

$2 (8 y^{3} + 3 {(2 y)}^{2} \cdot 1 + 3 (2 y) \cdot 1^{2} + 1^{3})$

Step 4.1.3

Apply the product rule to $2 y$ .

$2 (8 y^{3} + 3 (2^{2} y^{2}) \cdot 1 + 3 (2 y) \cdot 1^{2} + 1^{3})$

Step 4.1.4

Raise $2$ to the power of $2$ .

$2 (8 y^{3} + 3 (4 y^{2}) \cdot 1 + 3 (2 y) \cdot 1^{2} + 1^{3})$

Step 4.1.5

Multiply $4$ by $3$ .

$2 (8 y^{3} + 12 y^{2} \cdot 1 + 3 (2 y) \cdot 1^{2} + 1^{3})$

Step 4.1.6

Multiply $12$ by $1$ .

$2 (8 y^{3} + 12 y^{2} + 3 (2 y) \cdot 1^{2} + 1^{3})$

Step 4.1.7

Multiply $2$ by $3$ .

$2 (8 y^{3} + 12 y^{2} + 6 y \cdot 1^{2} + 1^{3})$

Step 4.1.8

One to any power is one.

$2 (8 y^{3} + 12 y^{2} + 6 y \cdot 1 + 1^{3})$

Step 4.1.9

Multiply $6$ by $1$ .

$2 (8 y^{3} + 12 y^{2} + 6 y + 1^{3})$

Step 4.1.10

One to any power is one.

$2 (8 y^{3} + 12 y^{2} + 6 y + 1)$

Step 4.2

Apply the distributive property.

$2 (8 y^{3}) + 2 (12 y^{2}) + 2 (6 y) + 2 \cdot 1$

Step 5

Simplify.

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Step 5.1

Multiply $8$ by $2$ .

$16 y^{3} + 2 (12 y^{2}) + 2 (6 y) + 2 \cdot 1$

Step 5.2

Multiply $12$ by $2$ .

$16 y^{3} + 24 y^{2} + 2 (6 y) + 2 \cdot 1$

Step 5.3

Multiply $6$ by $2$ .

$16 y^{3} + 24 y^{2} + 12 y + 2 \cdot 1$

Step 5.4

Multiply $2$ by $1$ .

$16 y^{3} + 24 y^{2} + 12 y + 2$