Simplify \( 5\sqrt{96} - 7\sqrt{54} \) - Math Solver AI

$Math Problem$

Mathematics High School

Question Content

Simplify \( 5\sqrt{96} - 7\sqrt{54} \)

Correct Answer

B

Detailed Solution Steps

1
Factor the radicands: \( \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6} \), \( \sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6} \)
2
Multiply by coefficients: \( 5\sqrt{96} = 5 \times 4\sqrt{6} = 20\sqrt{6} \), \( 7\sqrt{54} = 7 \times 3\sqrt{6} = 21\sqrt{6} \)
3
Subtract like radicals: \( 20\sqrt{6} - 21\sqrt{6} = -\sqrt{6} \)

Knowledge Points Involved

1

Simplifying Radicals

Breaking down the radicand into a product of a perfect square and another number, then using \( \sqrt{ab} = \sqrt{a}\sqrt{b} \) (a,b ≥ 0) to simplify.

2

Combining Like Radicals

Radicals with the same radicand can be combined by adding/subtracting their coefficients, similar to combining like terms in algebra.

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