We are given the equations:

1. $a \cdot b = 100$
2. $b \cdot c = 200$
3. $a \cdot c = 300$

We want to find $a + b + c$.

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Step 1: Multiply all three equations

$(a \cdot b) \cdot (b \cdot c) \cdot (a \cdot c) = 100 \cdot 200 \cdot 300$ $a^2 \cdot b^2 \cdot c^2 = 100 \cdot 200 \cdot 300$ Take the square root of both sides: $a \cdot b \cdot c = \sqrt{100 \cdot 200 \cdot 300}$ Simplify: $a \cdot b \cdot c = \sqrt{6,000,000} = 100\sqrt{6}$

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Step 2: Solve for $a, b, c$ individually

From the three equations:

• $b = \frac{100}{a}$
• $c = \frac{200}{b} = \frac{200}{\frac{100}{a}} = \frac{200a}{100} = 2a$
• Substitute $c = 2a$ into $a \cdot c = 300$: $a \cdot 2a = 300$ $2a^2 = 300$ $a^2 = 150 \quad \Rightarrow \quad a = \sqrt{150} = 5\sqrt{6}$

Now substitute $a = 5\sqrt{6}$ into the expressions for $b$ and $c$:

• $b = \frac{100}{a} = \frac{100}{5\sqrt{6}} = \frac{20}{\sqrt{6}} = \frac{20\sqrt{6}}{6} = \frac{10\sqrt{6}}{3}$
• $c = 2a = 2 \cdot 5\sqrt{6} = 10\sqrt{6}$

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Step 3: Find $a + b + c$

$a + b + c = 5\sqrt{6} + \frac{10\sqrt{6}}{3} + 10\sqrt{6}$ Combine the terms by expressing them with a common denominator: $a + b + c = \frac{15\sqrt{6}}{3} + \frac{10\sqrt{6}}{3} + \frac{30\sqrt{6}}{3}$ $a + b + c = \frac{55\sqrt{6}}{3}$

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Final Answer:

$a + b + c = \frac{55\sqrt{6}}{3}$

Would you like a detailed explanation of any part?

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Related Questions:

1. How do we rationalize the denominator in fractions involving square roots?
2. How does multiplying equations simplify problems involving multiple variables?
3. What is the significance of square roots in solving systems of equations?
4. Can this solution be expressed as a decimal approximation?
5. How can this problem be solved using matrices or numerical methods?

Tip:

When working with equations involving multiple variables and products, look for patterns like multiplying or dividing equations to eliminate variables systematically.