Question

equivalent to $10\sqrt{5}$ equivalent to $5\sqrt{10}$ equivalent to $10\sqrt{10}$
$2\sqrt{250}$ (a) $5\sqrt{2} \cdot \sqrt{5}$ (b) $5\sqrt{5} + 5\sqrt{5}$ (c) $\sqrt{50} \cdot \sqrt{5}$ (d) $\sqrt{5} \cdot \sqrt{10} \cdot \sqrt{10}$ (e)
answer
equivalent to $10\sqrt{5}$
letters: $\square$, $\square$
equivalent to $5\sqrt{10}$
letters: $\square$, $\square$
equivalent to $10\sqrt{10}$
letters: $\square$

Explanation:

Answer:

Equivalent to \( 10\sqrt{5} \): B. \( 5\sqrt{2} \cdot \sqrt{5} \), C. \( 5\sqrt{5} + 5\sqrt{5} \)

Equivalent to \( 5\sqrt{10} \): (No matches from given, maybe typo? Wait, let's re - check) Wait, let's recalculate each option:

Step 1: Analyze Option A: \( 2\sqrt{250} \)

Simplify \( \sqrt{250}=\sqrt{25\times10} = 5\sqrt{10} \), so \( 2\sqrt{250}=2\times5\sqrt{10}=10\sqrt{10} \)

Step 2: Analyze Option B: \( 5\sqrt{2}\cdot\sqrt{5} \)

Using the property \( \sqrt{a}\cdot\sqrt{b}=\sqrt{ab} \), we have \( 5\sqrt{2\times5}=5\sqrt{10} \)? Wait, no, \( \sqrt{2}\times\sqrt{5}=\sqrt{10} \), so \( 5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \)? Wait, no, wait \( \sqrt{2}\times\sqrt{5}=\sqrt{10} \), so \( 5\sqrt{2}\times\sqrt{5} = 5\sqrt{10} \)? Wait, no, I made a mistake. Wait \( \sqrt{2}\times\sqrt{5}=\sqrt{10} \), so \( 5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \)? But the first column is \( 10\sqrt{5} \). Wait, let's recalculate:

Wait, \( 5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \)? No, \( \sqrt{2}\times\sqrt{5}=\sqrt{10} \), so \( 5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \). Wait, maybe I misread the option. Wait, option B: \( 5\sqrt{2}\cdot\sqrt{5} \). Wait, \( \sqrt{2}\times\sqrt{5}=\sqrt{10} \), so \( 5\sqrt{10} \). But the first target is \( 10\sqrt{5} \). Wait, maybe option B was \( 5\sqrt{5}\cdot\sqrt{2} \)? No, let's check option C: \( 5\sqrt{5}+5\sqrt{5} \)
Combine like terms: \( (5 + 5)\sqrt{5}=10\sqrt{5} \)

Step 3: Analyze Option D: \( \sqrt{50}\cdot\sqrt{5} \)

\( \sqrt{50}=\sqrt{25\times2}=5\sqrt{2} \), so \( \sqrt{50}\times\sqrt{5}=5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \) (same as option B's miscalculation earlier, no, \( \sqrt{2}\times\sqrt{5}=\sqrt{10} \), so \( 5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \))

Step 4: Analyze Option E: \( \sqrt{5}\cdot\sqrt{10}\cdot\sqrt{10} \)

\( \sqrt{10}\times\sqrt{10} = 10 \), so \( \sqrt{5}\times10 = 10\sqrt{5} \)

Let's start over:

For \( 10\sqrt{5} \):

• Option C: \( 5\sqrt{5}+5\sqrt{5}=(5 + 5)\sqrt{5}=10\sqrt{5} \)
• Option E: \( \sqrt{5}\cdot\sqrt{10}\cdot\sqrt{10}=\sqrt{5}\times(\sqrt{10}\times\sqrt{10})=\sqrt{5}\times10 = 10\sqrt{5} \)
• Option B: Wait, \( 5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \) (not \( 10\sqrt{5} \))
• Option D: \( \sqrt{50}\times\sqrt{5}=\sqrt{250}\times\sqrt{5} \)? No, \( \sqrt{50}=5\sqrt{2} \), \( 5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \)
• Option A: \( 2\sqrt{250}=2\times5\sqrt{10}=10\sqrt{10} \)

For \( 5\sqrt{10} \):

• Option B: \( 5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \)
• Option D: \( \sqrt{50}\times\sqrt{5}=5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \)

For \( 10\sqrt{10} \):

• Option A: \( 2\sqrt{250}=10\sqrt{10} \)

So correcting the earlier mistake:

• Equivalent to \( 10\sqrt{5} \): C, E
• Equivalent to \( 5\sqrt{10} \): B, D
• Equivalent to \( 10\sqrt{10} \): A

Let's verify:

• Option C: \( 5\sqrt{5}+5\sqrt{5} = 10\sqrt{5} \) (correct)
• Option E: \( \sqrt{5}\times\sqrt{10}\times\sqrt{10}=\sqrt{5}\times10 = 10\sqrt{5} \) (correct)
• Option B: \( 5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \) (correct)
• Option D: \( \sqrt{50}\times\sqrt{5}=5\sqrt{2}\times\sqrt{5}=5\sqrt{10} \) (correct)
• Option A: \( 2\sqrt{250}=2\times5\sqrt{10}=10\sqrt{10} \) (correct)

So the correct groupings are:

• Equivalent to \( 10\sqrt{5} \): C, E
• Equivalent to \( 5\sqrt{10} \): B, D
• Equivalent to \( 10\sqrt{10} \): A