$$ \frac{15+\sqrt{18}}{3}= $$ $$ \frac{4-\sqrt{46}}{2}= $$ nplex Fractions (Numerators) $$ \begin{array}{l}\frac{65-12 \sqrt{75}}{5}= \ \frac{5+\sqrt{18}}{\sqrt{3}}=\end{array} $$

Question

UpStudy AI · Accepted Answer

Sure, let's simplify each of the given fractions step by step.

### 1. Simplify $$ \frac{15 + \sqrt{18}}{3} $$

**Step 1:** Simplify $$ \sqrt{18} $$.

$$
\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
$$

**Step 2:** Substitute back into the fraction.

$$
\frac{15 + \sqrt{18}}{3} = \frac{15 + 3\sqrt{2}}{3}
$$

**Step 3:** Divide each term in the numerator by 3.

$$
\frac{15}{3} + \frac{3\sqrt{2}}{3} = 5 + \sqrt{2}
$$

**Final Answer:**

$$
\frac{15 + \sqrt{18}}{3} = 5 + \sqrt{2}
$$

---

### 2. Simplify $$ \frac{4 - \sqrt{46}}{2} $$

**Step 1:** Split the fraction.

$$
\frac{4}{2} - \frac{\sqrt{46}}{2} = 2 - \frac{\sqrt{46}}{2}
$$

**Note:** The expression $$ \sqrt{46} $$ cannot be simplified further since 46 has no square factors other than 1.

**Final Answer:**

$$
\frac{4 - \sqrt{46}}{2} = 2 - \frac{\sqrt{46}}{2}
$$

---

### 3. Simplify the Nplex Fractions (Numerators)

#### a. Simplify $$ \frac{65 - 12\sqrt{75}}{5} $$

**Step 1:** Simplify $$ \sqrt{75} $$.

$$
\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}
$$

**Step 2:** Substitute back into the fraction.

$$
\frac{65 - 12\sqrt{75}}{5} = \frac{65 - 12 \times 5\sqrt{3}}{5} = \frac{65 - 60\sqrt{3}}{5}
$$

**Step 3:** Divide each term in the numerator by 5.

$$
\frac{65}{5} - \frac{60\sqrt{3}}{5} = 13 - 12\sqrt{3}
$$

**Final Answer:**

$$
\frac{65 - 12\sqrt{75}}{5} = 13 - 12\sqrt{3}
$$

#### b. Simplify $$ \frac{5 + \sqrt{18}}{\sqrt{3}} $$

**Step 1:** Simplify $$ \sqrt{18} $$.

$$
\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
$$

**Step 2:** Substitute back into the fraction.

$$
\frac{5 + \sqrt{18}}{\sqrt{3}} = \frac{5 + 3\sqrt{2}}{\sqrt{3}}
$$

**Step 3:** Rationalize the denominator by multiplying numerator and denominator by $$ \sqrt{3} $$.

$$
\frac{5 + 3\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3} + 3\sqrt{6}}{3}
$$

**Final Answer:**

$$
\frac{5 + \sqrt{18}}{\sqrt{3}} = \frac{5\sqrt{3} + 3\sqrt{6}}{3}
$$

---

**Summary of Simplified Expressions:**

$$
\begin{align*}
\frac{15 + \sqrt{18}}{3} &= 5 + \sqrt{2} \
\frac{4 - \sqrt{46}}{2} &= 2 - \frac{\sqrt{46}}{2} \
\frac{65 - 12\sqrt{75}}{5} &= 13 - 12\sqrt{3} \
\frac{5 + \sqrt{18}}{\sqrt{3}} &= \frac{5\sqrt{3} + 3\sqrt{6}}{3}
\end{align*}
$$
$$ \begin{align*} \frac{15 + \sqrt{18}}{3} &= 5 + \sqrt{2} \ \frac{4 - \sqrt{46}}{2} &= 2 - \frac{\sqrt{46}}{2} \ \frac{65 - 12\sqrt{75}}{5} &= 13 - 12\sqrt{3} \ \frac{5 + \sqrt{18}}{\sqrt{3}} &= \frac{5\sqrt{3} + 3\sqrt{6}}{3} \end{align*} $$

nplex Fractions (Numerators)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions