Algebra Examples

$- \sqrt{{(y + 7)}^{16}}$

Step 1

Rewrite ${(y + 7)}^{16}$ as ${({(y + 7)}^{8})}^{2}$ .

$- \sqrt{{({(y + 7)}^{8})}^{2}}$

Step 2

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

$- {(y + 7)}^{8}$

Step 3

Use the Binomial Theorem.

$- (y^{8} + 8 y^{7} \cdot 7 + 28 y^{6} \cdot 7^{2} + 56 y^{5} \cdot 7^{3} + 70 y^{4} \cdot 7^{4} + 56 y^{3} \cdot 7^{5} + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $7$ by $8$ .

$- (y^{8} + 56 y^{7} + 28 y^{6} \cdot 7^{2} + 56 y^{5} \cdot 7^{3} + 70 y^{4} \cdot 7^{4} + 56 y^{3} \cdot 7^{5} + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.2

Raise $7$ to the power of $2$ .

$- (y^{8} + 56 y^{7} + 28 y^{6} \cdot 49 + 56 y^{5} \cdot 7^{3} + 70 y^{4} \cdot 7^{4} + 56 y^{3} \cdot 7^{5} + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.3

Multiply $49$ by $28$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 56 y^{5} \cdot 7^{3} + 70 y^{4} \cdot 7^{4} + 56 y^{3} \cdot 7^{5} + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.4

Raise $7$ to the power of $3$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 56 y^{5} \cdot 343 + 70 y^{4} \cdot 7^{4} + 56 y^{3} \cdot 7^{5} + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.5

Multiply $343$ by $56$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 70 y^{4} \cdot 7^{4} + 56 y^{3} \cdot 7^{5} + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.6

Raise $7$ to the power of $4$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 70 y^{4} \cdot 2401 + 56 y^{3} \cdot 7^{5} + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.7

Multiply $2401$ by $70$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 168070 y^{4} + 56 y^{3} \cdot 7^{5} + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.8

Raise $7$ to the power of $5$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 168070 y^{4} + 56 y^{3} \cdot 16807 + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.9

Multiply $16807$ by $56$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 168070 y^{4} + 941192 y^{3} + 28 y^{2} \cdot 7^{6} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.10

Raise $7$ to the power of $6$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 168070 y^{4} + 941192 y^{3} + 28 y^{2} \cdot 117649 + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.11

Multiply $117649$ by $28$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 168070 y^{4} + 941192 y^{3} + 3294172 y^{2} + 8 y \cdot 7^{7} + 7^{8})$

Step 4.1.12

Raise $7$ to the power of $7$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 168070 y^{4} + 941192 y^{3} + 3294172 y^{2} + 8 y \cdot 823543 + 7^{8})$

Step 4.1.13

Multiply $823543$ by $8$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 168070 y^{4} + 941192 y^{3} + 3294172 y^{2} + 6588344 y + 7^{8})$

Step 4.1.14

Raise $7$ to the power of $8$ .

$- (y^{8} + 56 y^{7} + 1372 y^{6} + 19208 y^{5} + 168070 y^{4} + 941192 y^{3} + 3294172 y^{2} + 6588344 y + 5764801)$

Step 4.2

Apply the distributive property.

$- y^{8} - (56 y^{7}) - (1372 y^{6}) - (19208 y^{5}) - (168070 y^{4}) - (941192 y^{3}) - (3294172 y^{2}) - (6588344 y) - 1 \cdot 5764801$

Step 5

Simplify.

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Step 5.1

Multiply $56$ by $- 1$ .

$- y^{8} - 56 y^{7} - (1372 y^{6}) - (19208 y^{5}) - (168070 y^{4}) - (941192 y^{3}) - (3294172 y^{2}) - (6588344 y) - 1 \cdot 5764801$

Step 5.2

Multiply $1372$ by $- 1$ .

$- y^{8} - 56 y^{7} - 1372 y^{6} - (19208 y^{5}) - (168070 y^{4}) - (941192 y^{3}) - (3294172 y^{2}) - (6588344 y) - 1 \cdot 5764801$

Step 5.3

Multiply $19208$ by $- 1$ .

$- y^{8} - 56 y^{7} - 1372 y^{6} - 19208 y^{5} - (168070 y^{4}) - (941192 y^{3}) - (3294172 y^{2}) - (6588344 y) - 1 \cdot 5764801$

Step 5.4

Multiply $168070$ by $- 1$ .

$- y^{8} - 56 y^{7} - 1372 y^{6} - 19208 y^{5} - 168070 y^{4} - (941192 y^{3}) - (3294172 y^{2}) - (6588344 y) - 1 \cdot 5764801$

Step 5.5

Multiply $941192$ by $- 1$ .

$- y^{8} - 56 y^{7} - 1372 y^{6} - 19208 y^{5} - 168070 y^{4} - 941192 y^{3} - (3294172 y^{2}) - (6588344 y) - 1 \cdot 5764801$

Step 5.6

Multiply $3294172$ by $- 1$ .

$- y^{8} - 56 y^{7} - 1372 y^{6} - 19208 y^{5} - 168070 y^{4} - 941192 y^{3} - 3294172 y^{2} - (6588344 y) - 1 \cdot 5764801$

Step 5.7

Multiply $6588344$ by $- 1$ .

$- y^{8} - 56 y^{7} - 1372 y^{6} - 19208 y^{5} - 168070 y^{4} - 941192 y^{3} - 3294172 y^{2} - 6588344 y - 1 \cdot 5764801$

Step 5.8

Multiply $- 1$ by $5764801$ .

$- y^{8} - 56 y^{7} - 1372 y^{6} - 19208 y^{5} - 168070 y^{4} - 941192 y^{3} - 3294172 y^{2} - 6588344 y - 5764801$