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When simplifying square roots, we aim to make the expression inside the root as simple as possible. The main tool we use is the Quotient Property of Radicals, which states that \(\text{鈭殅\frac{a}{b} = \frac{\text{鈭殅a}{\text{鈭殅b}\) for any non-negative numbers \(a\) and \(b\).

Let's dive into part (a) of the problem: \(\text{鈭殅\frac{121}{16}\). Using the Quotient Property of Radicals, we separate the fraction into two parts: the square root of the numerator and the square root of the denominator.

So: \(\text{鈭殅\frac{121}{16} = \frac{\text{鈭殅121}{\text{鈭殅16}\).

Now we simplify each term separately:

• The square root of 121 is 11 because \(11^2 = 121\).
• The square root of 16 is 4 because \(4^2 = 16\).

Combining them, we get: \(\frac{11}{4}\).

That's our simplified result. Remember, always use the Quotient Property when dealing with radicals and fractions.

Nth roots are a generalization of square roots. The nth root of a number \(x\), denoted \(\text{鈦库垰}x\), is a value \(y\) such that \(y^n = x\). For example, the cube root \( \text{鲁鈭殅x \) is a number \(y\) where \( y^3 = x\).

In part (b) of the problem, we have \(\text{鲁鈭殅\frac{16}{250}\).

We use the Quotient Property again:
\(\text{鲁鈭殅\frac{16}{250} = \frac{\text{鲁鈭殅16}{\text{鲁鈭殅250}\)

Next, we simplify each term inside the cube root:

• The cube root of 16 is 2 because \(2^3 = 8\).
• The cube root of 250 simplifies to \(\text{鲁鈭殅125\) because \(125 = 5^3\).

Combining them, we get: \(\frac{2}{5}\).

This approach works for any nth root, not just cube roots. Always break down the root and simplify each part separately.

To tackle radical expressions efficiently, a step-by-step approach is crucial. Let's consider part (c): \(\text{鈦粹垰}\frac{32}{162}\).

Step 1: Identify the types of roots involved. Here, we're working with fourth roots.

Step 2: Apply the Quotient Property:
\( \text{鈦粹垰}\frac{32}{162} = \frac{\text{鈦粹垰}32}{\text{鈦粹垰}162}\).

Now we simplify each term:

• \text{鈦粹垰}32: Find the fourth root of 32. Since 32 = 2^5, we look for factors that make it easier. The closest fourth power within 32 is 16, \( 2^4.\).
• \text{鈦粹垰}162: Recognize that 162 can be factored as 2 x 81.

By breaking down 32 and 162 into prime factors, we see:
\(\text{鈦粹垰}2^5 = 2\), and \(\text{鈦粹垰}81 = 3\).

Combining these results:
\(\frac{2}{3}\).

Each part requires methodical use of the Quotient Property and simplification of roots. Follow these steps for each radical expression, and this systematic approach will make even complex problems manageable.