Basic Math Examples

${(\sqrt{5} + 4 \sqrt{3})}^{2}$

Step 1

Rewrite ${(\sqrt{5} + 4 \sqrt{3})}^{2}$ as $(\sqrt{5} + 4 \sqrt{3}) (\sqrt{5} + 4 \sqrt{3})$ .

$(\sqrt{5} + 4 \sqrt{3}) (\sqrt{5} + 4 \sqrt{3})$

Step 2

Expand $(\sqrt{5} + 4 \sqrt{3}) (\sqrt{5} + 4 \sqrt{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{5} (\sqrt{5} + 4 \sqrt{3}) + 4 \sqrt{3} (\sqrt{5} + 4 \sqrt{3})$

Step 2.2

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} (4 \sqrt{3}) + 4 \sqrt{3} (\sqrt{5} + 4 \sqrt{3})$

Step 2.3

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} (4 \sqrt{3}) + 4 \sqrt{3} \sqrt{5} + 4 \sqrt{3} (4 \sqrt{3})$

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

Combine using the product rule for radicals.

$\sqrt{5 \cdot 5} + \sqrt{5} (4 \sqrt{3}) + 4 \sqrt{3} \sqrt{5} + 4 \sqrt{3} (4 \sqrt{3})$

Step 3.1.2

Multiply $5$ by $5$ .

$\sqrt{25} + \sqrt{5} (4 \sqrt{3}) + 4 \sqrt{3} \sqrt{5} + 4 \sqrt{3} (4 \sqrt{3})$

Step 3.1.3

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2}} + \sqrt{5} (4 \sqrt{3}) + 4 \sqrt{3} \sqrt{5} + 4 \sqrt{3} (4 \sqrt{3})$

Step 3.1.4

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

$5 + \sqrt{5} (4 \sqrt{3}) + 4 \sqrt{3} \sqrt{5} + 4 \sqrt{3} (4 \sqrt{3})$

Step 3.1.5

Multiply $\sqrt{5} (4 \sqrt{3})$ .

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Step 3.1.5.1

Combine using the product rule for radicals.

$5 + 4 \sqrt{5 \cdot 3} + 4 \sqrt{3} \sqrt{5} + 4 \sqrt{3} (4 \sqrt{3})$

Step 3.1.5.2

Multiply $5$ by $3$ .

$5 + 4 \sqrt{15} + 4 \sqrt{3} \sqrt{5} + 4 \sqrt{3} (4 \sqrt{3})$

Step 3.1.6

Multiply $4 \sqrt{3} \sqrt{5}$ .

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Step 3.1.6.1

Combine using the product rule for radicals.

$5 + 4 \sqrt{15} + 4 \sqrt{5 \cdot 3} + 4 \sqrt{3} (4 \sqrt{3})$

Step 3.1.6.2

Multiply $5$ by $3$ .

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 4 \sqrt{3} (4 \sqrt{3})$

Step 3.1.7

Multiply $4 \sqrt{3} (4 \sqrt{3})$ .

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Step 3.1.7.1

Multiply $4$ by $4$ .

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 \sqrt{3} \sqrt{3}$

Step 3.1.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 ({\sqrt{3}}^{1} \sqrt{3})$

Step 3.1.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 3.1.7.4

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

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 {\sqrt{3}}^{1 + 1}$

Step 3.1.7.5

Add $1$ and $1$ .

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 {\sqrt{3}}^{2}$

Step 3.1.8

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 {(3^{\frac{1}{2}})}^{2}$

Step 3.1.8.2

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

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 3.1.8.3

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

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 \cdot 3^{\frac{2}{2}}$

Step 3.1.8.4

Cancel the common factor of $2$ .

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Step 3.1.8.4.1

Cancel the common factor.

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 \cdot 3^{\frac{2}{2}}$

Step 3.1.8.4.2

Rewrite the expression.

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 \cdot 3^{1}$

Step 3.1.8.5

Evaluate the exponent.

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 16 \cdot 3$

Step 3.1.9

Multiply $16$ by $3$ .

$5 + 4 \sqrt{15} + 4 \sqrt{15} + 48$

Step 3.2

Add $5$ and $48$ .

$53 + 4 \sqrt{15} + 4 \sqrt{15}$

Step 3.3

Add $4 \sqrt{15}$ and $4 \sqrt{15}$ .

$53 + 8 \sqrt{15}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$53 + 8 \sqrt{15}$

Decimal Form:

$83.98386676 \dots$