Problem Analysis #22

To solve the problem, we are given a value:

$$100\sqrt{8}$$

and asked to express it in the form:

$$\frac{p\sqrt{q}}{r}$$

where:

• $p, q, r$ are positive integers,
• $q$ is squarefree, and
• $\gcd(p, r) = 1$.

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Step 1: Simplify the Square Root

We begin by simplifying $$\sqrt{8}$$. Since:

$$8 = 4 \times 2 = 2^2 \times 2$$

we can write:

$$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$$

Now substitute back into the original expression:

$$100\sqrt{8} = 100 \cdot 2\sqrt{2} = 200\sqrt{2}$$

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Step 2: Express in the Required Form

We are to write this as:

$$\frac{p\sqrt{q}}{r}$$

Since 200$$\sqrt{2}$$ is already in the form where the denominator is 1 (i.e., it's equal to $$\frac{200\sqrt{2}}{1}$$), we identify:

• $p = 200$
• $q = 2$
• $r = 1$

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Step 3: Verify Conditions

Let’s check the required conditions:

• $p, q, r$ are positive integers: ✔️ (200, 2, 1)
• $q$ is squarefree: ✔️ (2 has no repeated prime factors)
• $\gcd(p, r) = \gcd(200, 1) = 1$: ✔️ (since 1 is coprime with any integer)

All conditions are satisfied.

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Step 4: Compute $p + q + r$

$$p + q + r = 200 + 2 + 1 = \boxed{203}$$

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Final Answer

$$\boxed{203}$$