Algebra Examples

$\sqrt{{(2 x - y)}^{8}}$

Step 1

Rewrite ${(2 x - y)}^{8}$ as ${({(2 x - y)}^{4})}^{2}$ .

$\sqrt{{({(2 x - y)}^{4})}^{2}}$

Step 2

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

${(2 x - y)}^{4}$

Step 3

Use the Binomial Theorem.

${(2 x)}^{4} + 4 {(2 x)}^{3} (- y) + 6 {(2 x)}^{2} {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4

Simplify each term.

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Step 4.1

Apply the product rule to $2 x$ .

$2^{4} x^{4} + 4 {(2 x)}^{3} (- y) + 6 {(2 x)}^{2} {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.2

Raise $2$ to the power of $4$ .

$16 x^{4} + 4 {(2 x)}^{3} (- y) + 6 {(2 x)}^{2} {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.3

Rewrite using the commutative property of multiplication.

$16 x^{4} + 4 \cdot - 1 {(2 x)}^{3} y + 6 {(2 x)}^{2} {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.4

Multiply $4$ by $- 1$ .

$16 x^{4} - 4 {(2 x)}^{3} y + 6 {(2 x)}^{2} {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.5

Apply the product rule to $2 x$ .

$16 x^{4} - 4 (2^{3} x^{3}) y + 6 {(2 x)}^{2} {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.6

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

$16 x^{4} - 4 (8 x^{3}) y + 6 {(2 x)}^{2} {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.7

Multiply $8$ by $- 4$ .

$16 x^{4} - 32 x^{3} y + 6 {(2 x)}^{2} {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.8

Apply the product rule to $2 x$ .

$16 x^{4} - 32 x^{3} y + 6 (2^{2} x^{2}) {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.9

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

$16 x^{4} - 32 x^{3} y + 6 (4 x^{2}) {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.10

Multiply $4$ by $6$ .

$16 x^{4} - 32 x^{3} y + 24 x^{2} {(- y)}^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.11

Apply the product rule to $- y$ .

$16 x^{4} - 32 x^{3} y + 24 x^{2} ({(- 1)}^{2} y^{2}) + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.12

Rewrite using the commutative property of multiplication.

$16 x^{4} - 32 x^{3} y + 24 \cdot {(- 1)}^{2} x^{2} y^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.13

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

$16 x^{4} - 32 x^{3} y + 24 \cdot 1 x^{2} y^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.14

Multiply $24$ by $1$ .

$16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} + 4 (2 x) {(- y)}^{3} + {(- y)}^{4}$

Step 4.15

Multiply $2$ by $4$ .

$16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} + 8 x {(- y)}^{3} + {(- y)}^{4}$

Step 4.16

Apply the product rule to $- y$ .

$16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} + 8 x ({(- 1)}^{3} y^{3}) + {(- y)}^{4}$

Step 4.17

Rewrite using the commutative property of multiplication.

$16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} + 8 \cdot {(- 1)}^{3} x y^{3} + {(- y)}^{4}$

Step 4.18

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

$16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} + 8 \cdot - 1 x y^{3} + {(- y)}^{4}$

Step 4.19

Multiply $8$ by $- 1$ .

$16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} - 8 x y^{3} + {(- y)}^{4}$

Step 4.20

Apply the product rule to $- y$ .

$16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} - 8 x y^{3} + {(- 1)}^{4} y^{4}$

Step 4.21

Raise $- 1$ to the power of $4$ .

$16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} - 8 x y^{3} + 1 y^{4}$

Step 4.22

Multiply $y^{4}$ by $1$ .

$16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} - 8 x y^{3} + y^{4}$