Simplify (3+ square root of 7)(5+3 square root of 28)

$(3 + \sqrt{7}) (5 + 3 \sqrt{28})$

Simplify each term.

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$(3 + \sqrt{7}) (5 + 3 \sqrt{4 (7)})$

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

$(3 + \sqrt{7}) (5 + 3 \sqrt{2^{2} \cdot 7})$

Pull terms out from under the radical.

$(3 + \sqrt{7}) (5 + 3 (2 \sqrt{7}))$

Multiply $2$ by $3$ .

$(3 + \sqrt{7}) (5 + 6 \sqrt{7})$

Expand $(3 + \sqrt{7}) (5 + 6 \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

$3 (5 + 6 \sqrt{7}) + \sqrt{7} (5 + 6 \sqrt{7})$

Apply the distributive property.

$3 \cdot 5 + 3 (6 \sqrt{7}) + \sqrt{7} (5 + 6 \sqrt{7})$

Apply the distributive property.

$3 \cdot 5 + 3 (6 \sqrt{7}) + \sqrt{7} \cdot 5 + \sqrt{7} (6 \sqrt{7})$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $5$ .

$15 + 3 (6 \sqrt{7}) + \sqrt{7} \cdot 5 + \sqrt{7} (6 \sqrt{7})$

Multiply $6$ by $3$ .

$15 + 18 \sqrt{7} + \sqrt{7} \cdot 5 + \sqrt{7} (6 \sqrt{7})$

Move $5$ to the left of $\sqrt{7}$ .

$15 + 18 \sqrt{7} + 5 \cdot \sqrt{7} + \sqrt{7} (6 \sqrt{7})$

Multiply $\sqrt{7} (6 \sqrt{7})$ .

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

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 ({\sqrt{7}}^{1} \sqrt{7})$

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

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})$

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

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 {\sqrt{7}}^{1 + 1}$

Add $1$ and $1$ .

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 {\sqrt{7}}^{2}$

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

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

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 {(7^{\frac{1}{2}})}^{2}$

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

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 \cdot 7^{\frac{1}{2} \cdot 2}$

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

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 \cdot 7^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 \cdot 7^{\frac{2}{2}}$

Rewrite the expression.

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 \cdot 7^{1}$

Evaluate the exponent.

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 6 \cdot 7$

Multiply $6$ by $7$ .

$15 + 18 \sqrt{7} + 5 \sqrt{7} + 42$

Add $15$ and $42$ .

$57 + 18 \sqrt{7} + 5 \sqrt{7}$

Add $18 \sqrt{7}$ and $5 \sqrt{7}$ .

$57 + 23 \sqrt{7}$

The result can be shown in multiple forms.

Exact Form:

$57 + 23 \sqrt{7}$

Decimal Form:

$117.85228015 \dots$

Simplify (3+ square root of 7)(5+3 square root of 28)

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