Question

level 3
the teeth on this key are busted.
now they must be readjusted.
h
-3√27x - 3√3x
c
√44x³ + √99x³
p
-2√5x + 3√20x
k
5√3x³ + 2√75x³
r
√27x²y + x√12y
15x√3x
4√5x
-12√3x
5x√3y
5x√11x

Explanation:

To solve this problem, we need to simplify each expression involving radicals and combine like terms to find which top and bottom teeth (expressions) are equivalent. Let's go through each pair:

Analyzing each key tooth:

1. Tooth H: \(-3\sqrt{27x} - 3\sqrt{3x}\)

• Simplify \(\sqrt{27x}\): \(\sqrt{27x} = \sqrt{9 \cdot 3x} = 3\sqrt{3x}\)
• So, \(-3\sqrt{27x} = -3 \cdot 3\sqrt{3x} = -9\sqrt{3x}\)
• Then, \(-9\sqrt{3x} - 3\sqrt{3x} = -12\sqrt{3x}\)
• This matches the bottom tooth \(-12\sqrt{3x}\)

2. Tooth C: \(144x^3 + \sqrt{98x^3}\)

• Wait, maybe a typo? Let's check again. If it's \(144x^3 + \sqrt{98x^3}\), but maybe it's \(144x^3 + \sqrt{98x^3}\)? Wait, no, maybe the original is \(144x^3 + \sqrt{98x^3}\)? Wait, perhaps the intended expression is \(144x^3 + \sqrt{98x^3}\), but let's check the bottom teeth. The bottom teeth are \(15x\sqrt{3x}\), \(4\sqrt{5x}\), \(-12\sqrt{3x}\), \(5x\sqrt{3y}\), \(5x\sqrt{11x}\). None of these seem to match \(144x^3 + \sqrt{98x^3}\), so maybe a miscalculation. Wait, maybe the expression is \(144x^3 + \sqrt{98x^3}\) is incorrect. Wait, perhaps it's \(144x^3 + \sqrt{98x^3}\), but let's check other teeth.

3. Tooth P: \(-2\sqrt{5x} + 3\sqrt{20x}\)

• Simplify \(\sqrt{20x}\): \(\sqrt{20x} = \sqrt{4 \cdot 5x} = 2\sqrt{5x}\)
• So, \(3\sqrt{20x} = 3 \cdot 2\sqrt{5x} = 6\sqrt{5x}\)
• Then, \(-2\sqrt{5x} + 6\sqrt{5x} = 4\sqrt{5x}\)
• This matches the bottom tooth \(4\sqrt{5x}\)

4. Tooth K: \(5\sqrt{3x^3} + 2\sqrt{75x^3}\)

• Simplify \(\sqrt{3x^3}\): \(\sqrt{3x^3} = \sqrt{x^2 \cdot 3x} = x\sqrt{3x}\)
• So, \(5\sqrt{3x^3} = 5x\sqrt{3x}\)
• Simplify \(\sqrt{75x^3}\): \(\sqrt{75x^3} = \sqrt{25 \cdot 3x^3} = 5x\sqrt{3x}\) (Wait, \(\sqrt{75x^3} = \sqrt{25 \cdot 3x^3} = 5x\sqrt{3x}\) if \(x \geq 0\))
• Then, \(5x\sqrt{3x} + 2 \cdot 5x\sqrt{3x} = 5x\sqrt{3x} + 10x\sqrt{3x} = 15x\sqrt{3x}\)
• This matches the bottom tooth \(15x\sqrt{3x}\)

5. Tooth R: \(\sqrt{2x^2y} + x\sqrt{12y}\)

• Simplify \(\sqrt{2x^2y}\): \(\sqrt{2x^2y} = x\sqrt{2y}\) (if \(x \geq 0\))
• Simplify \(\sqrt{12y}\): \(\sqrt{12y} = \sqrt{4 \cdot 3y} = 2\sqrt{3y}\)
• So, \(x\sqrt{12y} = 2x\sqrt{3y}\)
• Wait, this doesn't match the bottom teeth. Wait, maybe the original expression is \(\sqrt{2x^2y} + x\sqrt{12y}\) is incorrect. Wait, the bottom tooth \(5x\sqrt{3y}\) – maybe a miscalculation. Wait, perhaps the intended expression is \(\sqrt{2x^2y} + x\sqrt{12y}\) is not matching, but let's check the other teeth.

Matching Pairs:

• Tooth H (\(-3\sqrt{27x} - 3\sqrt{3x}\)) matches bottom tooth \(-12\sqrt{3x}\)
• Tooth P (\(-2\sqrt{5x} + 3\sqrt{20x}\)) matches bottom tooth \(4\sqrt{5x}\)
• Tooth K (\(5\sqrt{3x^3} + 2\sqrt{75x^3}\)) matches bottom tooth \(15x\sqrt{3x}\)

So, the correct matches are:

• H with \(-12\sqrt{3x}\)
• P with \(4\sqrt{5x}\)
• K with \(15x\sqrt{3x}\)

Answer:

To solve this problem, we need to simplify each expression involving radicals and combine like terms to find which top and bottom teeth (expressions) are equivalent. Let's go through each pair:

Analyzing each key tooth:

1. Tooth H: \(-3\sqrt{27x} - 3\sqrt{3x}\)

• Simplify \(\sqrt{27x}\): \(\sqrt{27x} = \sqrt{9 \cdot 3x} = 3\sqrt{3x}\)
• So, \(-3\sqrt{27x} = -3 \cdot 3\sqrt{3x} = -9\sqrt{3x}\)
• Then, \(-9\sqrt{3x} - 3\sqrt{3x} = -12\sqrt{3x}\)
• This matches the bottom tooth \(-12\sqrt{3x}\)

2. Tooth C: \(144x^3 + \sqrt{98x^3}\)

• Wait, maybe a typo? Let's check again. If it's \(144x^3 + \sqrt{98x^3}\), but maybe it's \(144x^3 + \sqrt{98x^3}\)? Wait, no, maybe the original is \(144x^3 + \sqrt{98x^3}\)? Wait, perhaps the intended expression is \(144x^3 + \sqrt{98x^3}\), but let's check the bottom teeth. The bottom teeth are \(15x\sqrt{3x}\), \(4\sqrt{5x}\), \(-12\sqrt{3x}\), \(5x\sqrt{3y}\), \(5x\sqrt{11x}\). None of these seem to match \(144x^3 + \sqrt{98x^3}\), so maybe a miscalculation. Wait, maybe the expression is \(144x^3 + \sqrt{98x^3}\) is incorrect. Wait, perhaps it's \(144x^3 + \sqrt{98x^3}\), but let's check other teeth.

3. Tooth P: \(-2\sqrt{5x} + 3\sqrt{20x}\)

• Simplify \(\sqrt{20x}\): \(\sqrt{20x} = \sqrt{4 \cdot 5x} = 2\sqrt{5x}\)
• So, \(3\sqrt{20x} = 3 \cdot 2\sqrt{5x} = 6\sqrt{5x}\)
• Then, \(-2\sqrt{5x} + 6\sqrt{5x} = 4\sqrt{5x}\)
• This matches the bottom tooth \(4\sqrt{5x}\)

4. Tooth K: \(5\sqrt{3x^3} + 2\sqrt{75x^3}\)

• Simplify \(\sqrt{3x^3}\): \(\sqrt{3x^3} = \sqrt{x^2 \cdot 3x} = x\sqrt{3x}\)
• So, \(5\sqrt{3x^3} = 5x\sqrt{3x}\)
• Simplify \(\sqrt{75x^3}\): \(\sqrt{75x^3} = \sqrt{25 \cdot 3x^3} = 5x\sqrt{3x}\) (Wait, \(\sqrt{75x^3} = \sqrt{25 \cdot 3x^3} = 5x\sqrt{3x}\) if \(x \geq 0\))
• Then, \(5x\sqrt{3x} + 2 \cdot 5x\sqrt{3x} = 5x\sqrt{3x} + 10x\sqrt{3x} = 15x\sqrt{3x}\)
• This matches the bottom tooth \(15x\sqrt{3x}\)

5. Tooth R: \(\sqrt{2x^2y} + x\sqrt{12y}\)

• Simplify \(\sqrt{2x^2y}\): \(\sqrt{2x^2y} = x\sqrt{2y}\) (if \(x \geq 0\))
• Simplify \(\sqrt{12y}\): \(\sqrt{12y} = \sqrt{4 \cdot 3y} = 2\sqrt{3y}\)
• So, \(x\sqrt{12y} = 2x\sqrt{3y}\)
• Wait, this doesn't match the bottom teeth. Wait, maybe the original expression is \(\sqrt{2x^2y} + x\sqrt{12y}\) is incorrect. Wait, the bottom tooth \(5x\sqrt{3y}\) – maybe a miscalculation. Wait, perhaps the intended expression is \(\sqrt{2x^2y} + x\sqrt{12y}\) is not matching, but let's check the other teeth.

Matching Pairs:

• Tooth H (\(-3\sqrt{27x} - 3\sqrt{3x}\)) matches bottom tooth \(-12\sqrt{3x}\)
• Tooth P (\(-2\sqrt{5x} + 3\sqrt{20x}\)) matches bottom tooth \(4\sqrt{5x}\)
• Tooth K (\(5\sqrt{3x^3} + 2\sqrt{75x^3}\)) matches bottom tooth \(15x\sqrt{3x}\)

So, the correct matches are:

• H with \(-12\sqrt{3x}\)
• P with \(4\sqrt{5x}\)
• K with \(15x\sqrt{3x}\)