Basic Math Examples

${(\sqrt{b} + 2 \sqrt{2})}^{2}$

Step 1

Rewrite ${(\sqrt{b} + 2 \sqrt{2})}^{2}$ as $(\sqrt{b} + 2 \sqrt{2}) (\sqrt{b} + 2 \sqrt{2})$ .

$(\sqrt{b} + 2 \sqrt{2}) (\sqrt{b} + 2 \sqrt{2})$

Step 2

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

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Step 2.1

Apply the distributive property.

$\sqrt{b} (\sqrt{b} + 2 \sqrt{2}) + 2 \sqrt{2} (\sqrt{b} + 2 \sqrt{2})$

Step 2.2

Apply the distributive property.

$\sqrt{b} \sqrt{b} + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} (\sqrt{b} + 2 \sqrt{2})$

Step 2.3

Apply the distributive property.

$\sqrt{b} \sqrt{b} + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

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 $\sqrt{b} \sqrt{b}$ .

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Step 3.1.1.1

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

${\sqrt{b}}^{1} \sqrt{b} + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

Step 3.1.1.2

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

${\sqrt{b}}^{1} {\sqrt{b}}^{1} + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

Step 3.1.1.3

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

${\sqrt{b}}^{1 + 1} + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

Step 3.1.1.4

Add $1$ and $1$ .

${\sqrt{b}}^{2} + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

$b^{\frac{2}{2}} + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

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.

$b^{\frac{2}{2}} + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

Step 3.1.2.4.2

Rewrite the expression.

$b^{1} + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

Step 3.1.2.5

Simplify.

$b + \sqrt{b} (2 \sqrt{2}) + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

Step 3.1.3

Combine using the product rule for radicals.

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{2} \sqrt{b} + 2 \sqrt{2} (2 \sqrt{2})$

Step 3.1.4

Combine using the product rule for radicals.

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 2 \sqrt{2} (2 \sqrt{2})$

Step 3.1.5

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

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Step 3.1.5.1

Multiply $2$ by $2$ .

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 \sqrt{2} \sqrt{2}$

Step 3.1.5.2

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

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 ({\sqrt{2}}^{1} \sqrt{2})$

Step 3.1.5.3

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

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 3.1.5.4

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

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 {\sqrt{2}}^{1 + 1}$

Step 3.1.5.5

Add $1$ and $1$ .

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 {\sqrt{2}}^{2}$

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

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 {(2^{\frac{1}{2}})}^{2}$

Step 3.1.6.2

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

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 3.1.6.3

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

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 \cdot 2^{\frac{2}{2}}$

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.

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 \cdot 2^{\frac{2}{2}}$

Step 3.1.6.4.2

Rewrite the expression.

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 \cdot 2^{1}$

Step 3.1.6.5

Evaluate the exponent.

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 4 \cdot 2$

Step 3.1.7

Multiply $4$ by $2$ .

$b + 2 \sqrt{b \cdot 2} + 2 \sqrt{b \cdot 2} + 8$

Step 3.2

Add $2 \sqrt{b \cdot 2}$ and $2 \sqrt{b \cdot 2}$ .

$b + 4 \sqrt{b \cdot 2} + 8$

Step 4

Reorder factors in $b + 4 \sqrt{b \cdot 2} + 8$ .

$b + 4 \sqrt{2 b} + 8$