# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-9/8

Exercises: Try Out - Set 9: Real exam simulation to sharpen your skills and build confidence. - Problem 8

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## Exercise 8

### Question

export const metadata = {
  title: "Problem 8",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

<MathContainer>
  <BlockMath math="\frac{(8 + \sqrt{5}) + (2\sqrt{5} + 1)}{\sqrt{5} + 3} = \dots" />
</MathContainer>


### Choices

- [ ] $$2$$
- [x] $$3$$
- [ ] $$5$$
- [ ] $$10$$
- [ ] $$14$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 8",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

#### Simplifying Algebraic Expression

We simplify the numerator by grouping like terms (integers with integers, radicals with radicals).

<MathContainer>
  <BlockMath math="\begin{aligned} \frac{(8 + \sqrt{5}) + (2\sqrt{5} + 1)}{\sqrt{5} + 3} &= \frac{(8 + 1) + (\sqrt{5} + 2\sqrt{5})}{\sqrt{5} + 3} \\ &= \frac{9 + 3\sqrt{5}}{\sqrt{5} + 3} \end{aligned}" />
</MathContainer>

Next, we factor the numerator by taking out the number <InlineMath math="3" />:

<MathContainer>
  <BlockMath math="\begin{aligned} \frac{9 + 3\sqrt{5}}{\sqrt{5} + 3} &= \frac{3(3 + \sqrt{5})}{\sqrt{5} + 3} \end{aligned}" />
</MathContainer>

Notice that <InlineMath math="(3 + \sqrt{5})" /> is equal to <InlineMath math="(\sqrt{5} + 3)" />. Thus, we can divide both the numerator and the denominator by this value:

<MathContainer>
  <BlockMath math="\frac{3(3 + \sqrt{5})}{\sqrt{5} + 3} = 3" />
</MathContainer>

Therefore, the final result is <InlineMath math="3" />.


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