Simplify fifth root of -(y-z)^15

$\sqrt[5]{- {(y - z)}^{15}}$

Step 1

Rewrite $- {(y - z)}^{15}$ as ${(- {(y - z)}^{3})}^{5}$ .

$\sqrt[5]{{(- {(y - z)}^{3})}^{5}}$

Step 2

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

$- {(y - z)}^{3}$

Step 3

Use the Binomial Theorem.

$- (y^{3} + 3 y^{2} (- z) + 3 y {(- z)}^{2} + {(- z)}^{3})$

Step 4

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $3$ by $- 1$ .

$- (y^{3} - 3 y^{2} z + 3 y {(- z)}^{2} + {(- z)}^{3})$

Apply the product rule to $- z$ .

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

Rewrite using the commutative property of multiplication.

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

Raise $- 1$ to the power of $2$ .

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

Multiply $3$ by $1$ .

$- (y^{3} - 3 y^{2} z + 3 y z^{2} + {(- z)}^{3})$

Apply the product rule to $- z$ .

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

Raise $- 1$ to the power of $3$ .

$- (y^{3} - 3 y^{2} z + 3 y z^{2} - z^{3})$

Apply the distributive property.

$- y^{3} - (- 3 y^{2} z) - (3 y z^{2}) - - z^{3}$

Step 5

Simplify.

Multiply $- 3$ by $- 1$ .

$- y^{3} + 3 (y^{2} z) - (3 y z^{2}) - - z^{3}$

Multiply $3$ by $- 1$ .

$- y^{3} + 3 (y^{2} z) - 3 (y z^{2}) - - z^{3}$

Multiply $- - z^{3}$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $z^{3}$ by $1$ .

$- y^{3} + 3 (y^{2} z) - 3 (y z^{2}) + z^{3}$

Step 6

Remove parentheses.

$- y^{3} + 3 y^{2} z - 3 y z^{2} + z^{3}$