Evaluate ( square root of 3+4 square root of 7)^2

${(\sqrt{3} + 4 \sqrt{7})}^{2}$

Step 1

Rewrite ${(\sqrt{3} + 4 \sqrt{7})}^{2}$ as $(\sqrt{3} + 4 \sqrt{7}) (\sqrt{3} + 4 \sqrt{7})$ .

$(\sqrt{3} + 4 \sqrt{7}) (\sqrt{3} + 4 \sqrt{7})$

Step 2

Expand $(\sqrt{3} + 4 \sqrt{7}) (\sqrt{3} + 4 \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{3} (\sqrt{3} + 4 \sqrt{7}) + 4 \sqrt{7} (\sqrt{3} + 4 \sqrt{7})$

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (4 \sqrt{7}) + 4 \sqrt{7} (\sqrt{3} + 4 \sqrt{7})$

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (4 \sqrt{7}) + 4 \sqrt{7} \sqrt{3} + 4 \sqrt{7} (4 \sqrt{7})$

Step 3

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\sqrt{3 \cdot 3} + \sqrt{3} (4 \sqrt{7}) + 4 \sqrt{7} \sqrt{3} + 4 \sqrt{7} (4 \sqrt{7})$

Multiply $3$ by $3$ .

$\sqrt{9} + \sqrt{3} (4 \sqrt{7}) + 4 \sqrt{7} \sqrt{3} + 4 \sqrt{7} (4 \sqrt{7})$

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}} + \sqrt{3} (4 \sqrt{7}) + 4 \sqrt{7} \sqrt{3} + 4 \sqrt{7} (4 \sqrt{7})$

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

$3 + \sqrt{3} (4 \sqrt{7}) + 4 \sqrt{7} \sqrt{3} + 4 \sqrt{7} (4 \sqrt{7})$

Multiply $\sqrt{3} (4 \sqrt{7})$ .

Combine using the product rule for radicals.

$3 + 4 \sqrt{3 \cdot 7} + 4 \sqrt{7} \sqrt{3} + 4 \sqrt{7} (4 \sqrt{7})$

Multiply $3$ by $7$ .

$3 + 4 \sqrt{21} + 4 \sqrt{7} \sqrt{3} + 4 \sqrt{7} (4 \sqrt{7})$

Multiply $4 \sqrt{7} \sqrt{3}$ .

Combine using the product rule for radicals.

$3 + 4 \sqrt{21} + 4 \sqrt{3 \cdot 7} + 4 \sqrt{7} (4 \sqrt{7})$

Multiply $3$ by $7$ .

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 4 \sqrt{7} (4 \sqrt{7})$

Multiply $4 \sqrt{7} (4 \sqrt{7})$ .

Multiply $4$ by $4$ .

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 \sqrt{7} \sqrt{7}$

Raise $\sqrt{7}$ to the power of $1$ .

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 ({\sqrt{7}}^{1} \sqrt{7})$

Raise $\sqrt{7}$ to the power of $1$ .

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})$

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

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 {\sqrt{7}}^{1 + 1}$

Add $1$ and $1$ .

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 {\sqrt{7}}^{2}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 {(7^{\frac{1}{2}})}^{2}$

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

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 \cdot 7^{\frac{1}{2} \cdot 2}$

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

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 \cdot 7^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 \cdot 7^{\frac{2}{2}}$

Rewrite the expression.

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 \cdot 7^{1}$

Evaluate the exponent.

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 16 \cdot 7$

Multiply $16$ by $7$ .

$3 + 4 \sqrt{21} + 4 \sqrt{21} + 112$

Add $3$ and $112$ .

$115 + 4 \sqrt{21} + 4 \sqrt{21}$

Add $4 \sqrt{21}$ and $4 \sqrt{21}$ .

$115 + 8 \sqrt{21}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$115 + 8 \sqrt{21}$

Decimal Form:

$151.66060555 \dots$

Evaluate ( square root of 3+4 square root of 7)^2

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