Simplify ( square root of 7- square root of 2)^2

${(\sqrt{7} - \sqrt{2})}^{2}$

Step 1

Rewrite ${(\sqrt{7} - \sqrt{2})}^{2}$ as $(\sqrt{7} - \sqrt{2}) (\sqrt{7} - \sqrt{2})$ .

$(\sqrt{7} - \sqrt{2}) (\sqrt{7} - \sqrt{2})$

Step 2

Expand $(\sqrt{7} - \sqrt{2}) (\sqrt{7} - \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{7} (\sqrt{7} - \sqrt{2}) - \sqrt{2} (\sqrt{7} - \sqrt{2})$

Apply the distributive property.

$\sqrt{7} \sqrt{7} + \sqrt{7} (- \sqrt{2}) - \sqrt{2} (\sqrt{7} - \sqrt{2})$

Apply the distributive property.

$\sqrt{7} \sqrt{7} + \sqrt{7} (- \sqrt{2}) - \sqrt{2} \sqrt{7} - \sqrt{2} (- \sqrt{2})$

Step 3

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\sqrt{7 \cdot 7} + \sqrt{7} (- \sqrt{2}) - \sqrt{2} \sqrt{7} - \sqrt{2} (- \sqrt{2})$

Multiply $7$ by $7$ .

$\sqrt{49} + \sqrt{7} (- \sqrt{2}) - \sqrt{2} \sqrt{7} - \sqrt{2} (- \sqrt{2})$

Rewrite $49$ as $7^{2}$ .

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

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

$7 + \sqrt{7} (- \sqrt{2}) - \sqrt{2} \sqrt{7} - \sqrt{2} (- \sqrt{2})$

Multiply $\sqrt{7} (- \sqrt{2})$ .

Combine using the product rule for radicals.

$7 - \sqrt{7 \cdot 2} - \sqrt{2} \sqrt{7} - \sqrt{2} (- \sqrt{2})$

Multiply $7$ by $2$ .

$7 - \sqrt{14} - \sqrt{2} \sqrt{7} - \sqrt{2} (- \sqrt{2})$

Multiply $- \sqrt{2} \sqrt{7}$ .

Combine using the product rule for radicals.

$7 - \sqrt{14} - \sqrt{7 \cdot 2} - \sqrt{2} (- \sqrt{2})$

Multiply $7$ by $2$ .

$7 - \sqrt{14} - \sqrt{14} - \sqrt{2} (- \sqrt{2})$

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

Multiply $- 1$ by $- 1$ .

$7 - \sqrt{14} - \sqrt{14} + 1 \sqrt{2} \sqrt{2}$

Multiply $\sqrt{2}$ by $1$ .

$7 - \sqrt{14} - \sqrt{14} + \sqrt{2} \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$7 - \sqrt{14} - \sqrt{14} + {\sqrt{2}}^{1} \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$7 - \sqrt{14} - \sqrt{14} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$7 - \sqrt{14} - \sqrt{14} + {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$7 - \sqrt{14} - \sqrt{14} + {\sqrt{2}}^{2}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$7 - \sqrt{14} - \sqrt{14} + {(2^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$7 - \sqrt{14} - \sqrt{14} + 2^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$7 - \sqrt{14} - \sqrt{14} + 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$7 - \sqrt{14} - \sqrt{14} + 2^{\frac{2}{2}}$

Rewrite the expression.

$7 - \sqrt{14} - \sqrt{14} + 2^{1}$

Evaluate the exponent.

$7 - \sqrt{14} - \sqrt{14} + 2$

Add $7$ and $2$ .

$9 - \sqrt{14} - \sqrt{14}$

Subtract $\sqrt{14}$ from $- \sqrt{14}$ .

$9 - 2 \sqrt{14}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$9 - 2 \sqrt{14}$

Decimal Form:

$1.51668522 \dots$

Simplify ( square root of 7- square root of 2)^2

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