Simplify ( square root of y- square root of 2)^2

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

Step 1

Rewrite

${(\sqrt{y} - \sqrt{2})}^{2}$ as

$(\sqrt{y} - \sqrt{2}) (\sqrt{y} - \sqrt{2})$ .

$(\sqrt{y} - \sqrt{2}) (\sqrt{y} - \sqrt{2})$

Step 2

Expand

$(\sqrt{y} - \sqrt{2}) (\sqrt{y} - \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{y} (\sqrt{y} - \sqrt{2}) - \sqrt{2} (\sqrt{y} - \sqrt{2})$

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply

$\sqrt{y} \sqrt{y}$ .

Raise

$\sqrt{y}$ to the power of

$1$ .

${\sqrt{y}}^{1} \sqrt{y} + \sqrt{y} (- \sqrt{2}) - \sqrt{2} \sqrt{y} - \sqrt{2} (- \sqrt{2})$

Raise

$\sqrt{y}$ to the power of

$1$ .

${\sqrt{y}}^{1} {\sqrt{y}}^{1} + \sqrt{y} (- \sqrt{2}) - \sqrt{2} \sqrt{y} - \sqrt{2} (- \sqrt{2})$

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\sqrt{y}}^{1 + 1} + \sqrt{y} (- \sqrt{2}) - \sqrt{2} \sqrt{y} - \sqrt{2} (- \sqrt{2})$

Add

$1$ and

$1$ .

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

Rewrite

${\sqrt{y}}^{2}$ as

$y$ .

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{y}$ as

$y^{\frac{1}{2}}$ .

${(y^{\frac{1}{2}})}^{2} + \sqrt{y} (- \sqrt{2}) - \sqrt{2} \sqrt{y} - \sqrt{2} (- \sqrt{2})$

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{2} \cdot 2} + \sqrt{y} (- \sqrt{2}) - \sqrt{2} \sqrt{y} - \sqrt{2} (- \sqrt{2})$

Combine

$\frac{1}{2}$ and

$2$ .

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

Cancel the common factor of

$2$ .

Cancel the common factor.

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

Rewrite the expression.

$y^{1} + \sqrt{y} (- \sqrt{2}) - \sqrt{2} \sqrt{y} - \sqrt{2} (- \sqrt{2})$

Simplify.

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

Combine using the product rule for radicals.

$y - \sqrt{y \cdot 2} - \sqrt{2} \sqrt{y} - \sqrt{2} (- \sqrt{2})$

Combine using the product rule for radicals.

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} - \sqrt{2} (- \sqrt{2})$

Multiply

$- \sqrt{2} (- \sqrt{2})$ .

Multiply

$- 1$ by

$- 1$ .

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + 1 \sqrt{2} \sqrt{2}$

Multiply

$\sqrt{2}$ by

$1$ .

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + \sqrt{2} \sqrt{2}$

Raise

$\sqrt{2}$ to the power of

$1$ .

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + {\sqrt{2}}^{1} \sqrt{2}$

Raise

$\sqrt{2}$ to the power of

$1$ .

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}$

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + {\sqrt{2}}^{1 + 1}$

Add

$1$ and

$1$ .

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + {\sqrt{2}}^{2}$

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + {(2^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + 2^{\frac{1}{2} \cdot 2}$

Combine

$\frac{1}{2}$ and

$2$ .

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + 2^{\frac{2}{2}}$

Cancel the common factor of

$2$ .

Cancel the common factor.

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + 2^{\frac{2}{2}}$

Rewrite the expression.

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + 2^{1}$

Evaluate the exponent.

$y - \sqrt{y \cdot 2} - \sqrt{y \cdot 2} + 2$

Subtract

$\sqrt{y \cdot 2}$ from

$- \sqrt{y \cdot 2}$ .

$y - 2 \sqrt{y \cdot 2} + 2$

Step 4

Reorder factors in

$y - 2 \sqrt{y \cdot 2} + 2$ .

$y - 2 \sqrt{2 y} + 2$

Simplify ( square root of y- square root of 2)^2

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