Simplify ( square root of x- square root of 8)( square root of x+ square root of 2)

$(\sqrt{x} - \sqrt{8}) (\sqrt{x} + \sqrt{2})$

Step 1

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$(\sqrt{x} - \sqrt{4 (2)}) (\sqrt{x} + \sqrt{2})$

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x} \sqrt{x}$ .

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

${\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} \sqrt{2} - 2 \sqrt{2} \sqrt{x} - 2 \sqrt{2} \sqrt{2}$

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

${\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} \sqrt{2} - 2 \sqrt{2} \sqrt{x} - 2 \sqrt{2} \sqrt{2}$

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

${\sqrt{x}}^{1 + 1} + \sqrt{x} \sqrt{2} - 2 \sqrt{2} \sqrt{x} - 2 \sqrt{2} \sqrt{2}$

Add $1$ and $1$ .

${\sqrt{x}}^{2} + \sqrt{x} \sqrt{2} - 2 \sqrt{2} \sqrt{x} - 2 \sqrt{2} \sqrt{2}$

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

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

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

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

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

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

$x^{\frac{2}{2}} + \sqrt{x} \sqrt{2} - 2 \sqrt{2} \sqrt{x} - 2 \sqrt{2} \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{2}{2}} + \sqrt{x} \sqrt{2} - 2 \sqrt{2} \sqrt{x} - 2 \sqrt{2} \sqrt{2}$

Rewrite the expression.

$x^{1} + \sqrt{x} \sqrt{2} - 2 \sqrt{2} \sqrt{x} - 2 \sqrt{2} \sqrt{2}$

Simplify.

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

Combine using the product rule for radicals.

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

Combine using the product rule for radicals.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 2$ by $2$ .

$x + \sqrt{x \cdot 2} - 2 \sqrt{x \cdot 2} - 4$

Subtract $2 \sqrt{x \cdot 2}$ from $\sqrt{x \cdot 2}$ .

$x - \sqrt{x \cdot 2} - 4$

Step 4

Reorder factors in $x - \sqrt{x \cdot 2} - 4$ .

$x - \sqrt{2 x} - 4$

Simplify ( square root of x- square root of 8)( square root of x+ square root of 2)

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