Algebra Examples

${(5 \sqrt{x} - \sqrt{2})}^{2}$

Step 1

Rewrite ${(5 \sqrt{x} - \sqrt{2})}^{2}$ as $(5 \sqrt{x} - \sqrt{2}) (5 \sqrt{x} - \sqrt{2})$ .

$(5 \sqrt{x} - \sqrt{2}) (5 \sqrt{x} - \sqrt{2})$

Step 2

Expand $(5 \sqrt{x} - \sqrt{2}) (5 \sqrt{x} - \sqrt{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$5 \sqrt{x} (5 \sqrt{x} - \sqrt{2}) - \sqrt{2} (5 \sqrt{x} - \sqrt{2})$

Step 2.2

Apply the distributive property.

$5 \sqrt{x} (5 \sqrt{x}) + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x} - \sqrt{2})$

Step 2.3

Apply the distributive property.

$5 \sqrt{x} (5 \sqrt{x}) + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $5 \sqrt{x} (5 \sqrt{x})$ .

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Step 3.1.1.1

Multiply $5$ by $5$ .

$25 \sqrt{x} \sqrt{x} + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.1.2

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

$25 ({\sqrt{x}}^{1} \sqrt{x}) + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.1.3

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

$25 ({\sqrt{x}}^{1} {\sqrt{x}}^{1}) + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.1.4

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

$25 {\sqrt{x}}^{1 + 1} + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.1.5

Add $1$ and $1$ .

$25 {\sqrt{x}}^{2} + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.2

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

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Step 3.1.2.1

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

$25 {(x^{\frac{1}{2}})}^{2} + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.2.2

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

$25 x^{\frac{1}{2} \cdot 2} + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.2.3

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

$25 x^{\frac{2}{2}} + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.1

Cancel the common factor.

$25 x^{\frac{2}{2}} + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.2.4.2

Rewrite the expression.

$25 x^{1} + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.2.5

Simplify.

$25 x + 5 \sqrt{x} (- \sqrt{2}) - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.3

Multiply $5 \sqrt{x} (- \sqrt{2})$ .

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Step 3.1.3.1

Multiply $- 1$ by $5$ .

$25 x - 5 \sqrt{x} \sqrt{2} - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.3.2

Combine using the product rule for radicals.

$25 x - 5 \sqrt{2 x} - \sqrt{2} (5 \sqrt{x}) - \sqrt{2} (- \sqrt{2})$

Step 3.1.4

Multiply $- \sqrt{2} (5 \sqrt{x})$ .

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Step 3.1.4.1

Multiply $5$ by $- 1$ .

$25 x - 5 \sqrt{2 x} - 5 \sqrt{2} \sqrt{x} - \sqrt{2} (- \sqrt{2})$

Step 3.1.4.2

Combine using the product rule for radicals.

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} - \sqrt{2} (- \sqrt{2})$

Step 3.1.5

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

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Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + 1 \sqrt{2} \sqrt{2}$

Step 3.1.5.2

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

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + \sqrt{2} \sqrt{2}$

Step 3.1.5.3

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

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + {\sqrt{2}}^{1} \sqrt{2}$

Step 3.1.5.4

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

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}$

Step 3.1.5.5

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

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + {\sqrt{2}}^{1 + 1}$

Step 3.1.5.6

Add $1$ and $1$ .

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + {\sqrt{2}}^{2}$

Step 3.1.6

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

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Step 3.1.6.1

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

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + {(2^{\frac{1}{2}})}^{2}$

Step 3.1.6.2

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

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + 2^{\frac{1}{2} \cdot 2}$

Step 3.1.6.3

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

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + 2^{\frac{2}{2}}$

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + 2^{\frac{2}{2}}$

Step 3.1.6.4.2

Rewrite the expression.

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + 2^{1}$

Step 3.1.6.5

Evaluate the exponent.

$25 x - 5 \sqrt{2 x} - 5 \sqrt{x \cdot 2} + 2$

Step 3.2

Subtract $5 \sqrt{x \cdot 2}$ from $- 5 \sqrt{2 x}$ .

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Step 3.2.1

Reorder $x$ and $2$ .

$25 x - 5 \sqrt{2 x} - 5 \sqrt{2 \cdot x} + 2$

Step 3.2.2

Subtract $5 \sqrt{2 \cdot x}$ from $- 5 \sqrt{2 x}$ .

$25 x - 10 \sqrt{2 x} + 2$