$$ N $$ that $$ \frac{\sqrt{12}-\sqrt{75}}{3^{\frac{3}{2}}}=-1 $$

Question

UpStudy AI · Accepted Answer

Let's evaluate the expression:

$$
\frac{\sqrt{12} - \sqrt{75}}{3^{\frac{3}{2}}}
$$

**Step 1: Simplify the Square Roots**

1. $$\sqrt{12}$$:
   $$
   \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}
   $$

2. $$\sqrt{75}$$:
   $$
   \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}
   $$

**Step 2: Substitute the Simplified Roots Back Into the Expression**

$$
\frac{\sqrt{12} - \sqrt{75}}{3^{\frac{3}{2}}} = \frac{2\sqrt{3} - 5\sqrt{3}}{3^{\frac{3}{2}}}
$$

**Step 3: Combine Like Terms in the Numerator**

$$
2\sqrt{3} - 5\sqrt{3} = (2 - 5)\sqrt{3} = -3\sqrt{3}
$$

So now, the expression is:

$$
\frac{-3\sqrt{3}}{3^{\frac{3}{2}}}
$$

**Step 4: Simplify the Denominator**

$$
3^{\frac{3}{2}} = 3^{1} \times 3^{\frac{1}{2}} = 3 \times \sqrt{3} = 3\sqrt{3}
$$

**Step 5: Substitute Back and Simplify the Fraction**

$$
\frac{-3\sqrt{3}}{3\sqrt{3}} = \frac{-3}{3} \times \frac{\sqrt{3}}{\sqrt{3}} = -1 \times 1 = -1
$$

**Conclusion:**

$$
\frac{\sqrt{12} - \sqrt{75}}{3^{\frac{3}{2}}} = -1
$$
$$ \frac{\sqrt{12} - \sqrt{75}}{3^{\frac{3}{2}}} = -1 $$

\( N \) that \( \frac{\sqrt{12}-\sqrt{75}}{3^{\frac{3}{2}}}=-1 \)