Simplify the Radical Expression sixth root of 64(2y+1)^18

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

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.

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

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 each term.

Apply the product rule to $2 y$ .

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

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

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

Apply the product rule to $2 y$ .

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

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

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

Multiply $4$ by $3$ .

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

Multiply $12$ by $1$ .

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

Multiply $2$ by $3$ .

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

One to any power is one.

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

Multiply $6$ by $1$ .

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

One to any power is one.

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

Step 5

Apply the distributive property.

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

Step 6

Simplify.

Multiply $8$ by $2$ .

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

Multiply $12$ by $2$ .

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

Multiply $6$ by $2$ .

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

Multiply $2$ by $1$ .

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