Basic Math Examples

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

Step 1

Factor using the perfect square rule.

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Step 1.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 z y = 2 \cdot z \cdot y$

Step 1.2

Rewrite the polynomial.

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

Step 1.3

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = y$ .

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

Step 2

Rewrite ${(z - y)}^{2} {(z + y)}^{2}$ as ${((z - y) (z + y))}^{2}$ .

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

Step 3

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

$(z - y) (z + y)$

Step 4

Expand $(z - y) (z + y)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$z (z + y) - y (z + y)$

Step 4.2

Apply the distributive property.

$z \cdot z + z y - y (z + y)$

Step 4.3

Apply the distributive property.

$z \cdot z + z y - y z - y \cdot y$

Step 5

Simplify terms.

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Step 5.1

Combine the opposite terms in $z \cdot z + z y - y z - y \cdot y$ .

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Step 5.1.1

Reorder the factors in the terms $z y$ and $- y z$ .

$z \cdot z + y z - y z - y \cdot y$

Step 5.1.2

Subtract $y z$ from $y z$ .

$z \cdot z + 0 - y \cdot y$

Step 5.1.3

Add $z \cdot z$ and $0$ .

$z \cdot z - y \cdot y$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $z$ by $z$ .

$z^{2} - y \cdot y$

Step 5.2.2

Multiply $y$ by $y$ by adding the exponents.

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Step 5.2.2.1

Move $y$ .

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

Step 5.2.2.2

Multiply $y$ by $y$ .

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