Question

simplify the following radical expression.

\\((1 + 2\sqrt{7})(3 + 4\sqrt{7})\\)

\\(? + \sqrt{ }\\)

Explanation:

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<reasoning>

Expand the expression using FOIL

\[
(1 + 2\sqrt{7})(3 + 4\sqrt{7}) = 1 \cdot 3 + 1 \cdot 4\sqrt{7} + 2\sqrt{7} \cdot 3 + 2\sqrt{7} \cdot 4\sqrt{7}
\]

Simplify individual terms

\[

$$\begin{aligned} 1 \cdot 3 &= 3 \\ 1 \cdot 4\sqrt{7} &= 4\sqrt{7} \\ 2\sqrt{7} \cdot 3 &= 6\sqrt{7} \\ 2\sqrt{7} \cdot 4\sqrt{7} &= 8 \cdot 7 = 56 \end{aligned}$$

\]

Combine like terms

\[

$$\begin{aligned} (3 + 56) + (4\sqrt{7} + 6\sqrt{7}) &= 59 + 10\sqrt{7} \end{aligned}$$

\]
</reasoning>

<answer>
Simplify the following radical expression.

\[(1 + 2\sqrt{7})(3 + 4\sqrt{7})\]

<blank>59</blank> + <blank>10</blank>\(\sqrt{\text{7}}\)
</answer>

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Answer:

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<reasoning>

Expand the expression using FOIL

\[
(1 + 2\sqrt{7})(3 + 4\sqrt{7}) = 1 \cdot 3 + 1 \cdot 4\sqrt{7} + 2\sqrt{7} \cdot 3 + 2\sqrt{7} \cdot 4\sqrt{7}
\]

Simplify individual terms

\[

$$\begin{aligned} 1 \cdot 3 &= 3 \\ 1 \cdot 4\sqrt{7} &= 4\sqrt{7} \\ 2\sqrt{7} \cdot 3 &= 6\sqrt{7} \\ 2\sqrt{7} \cdot 4\sqrt{7} &= 8 \cdot 7 = 56 \end{aligned}$$

\]

Combine like terms

\[

$$\begin{aligned} (3 + 56) + (4\sqrt{7} + 6\sqrt{7}) &= 59 + 10\sqrt{7} \end{aligned}$$

\]
</reasoning>

<answer>
Simplify the following radical expression.

\[(1 + 2\sqrt{7})(3 + 4\sqrt{7})\]

<blank>59</blank> + <blank>10</blank>\(\sqrt{\text{7}}\)
</answer>

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