What is the following product

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The Correct Answer and Explanation is:

We are asked to simplify the product: (12+6)(6−10)(\sqrt{12} + \sqrt{6})(\sqrt{6} – \sqrt{10})

Step 1: Simplify Square Roots

Let’s simplify any square roots where possible:

  • 12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
  • 6\sqrt{6} and 10\sqrt{10} are already simplified.

So the expression becomes: (23+6)(6−10)(2\sqrt{3} + \sqrt{6})(\sqrt{6} – \sqrt{10})

Step 2: Use FOIL (Distributive Property)

Apply the distributive property (FOIL): =(23)(6)+(6)(6)−(23)(10)−(6)(10)= (2\sqrt{3})(\sqrt{6}) + (\sqrt{6})(\sqrt{6}) – (2\sqrt{3})(\sqrt{10}) – (\sqrt{6})(\sqrt{10})

Let’s evaluate each term:

  1. 23⋅6=218=2⋅9⋅2=2⋅32=622\sqrt{3} \cdot \sqrt{6} = 2\sqrt{18} = 2 \cdot \sqrt{9 \cdot 2} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}
  2. 6⋅6=36=6\sqrt{6} \cdot \sqrt{6} = \sqrt{36} = 6
  3. −23⋅10=−230-2\sqrt{3} \cdot \sqrt{10} = -2\sqrt{30}
  4. −6⋅10=−60=−4⋅15=−215-\sqrt{6} \cdot \sqrt{10} = -\sqrt{60} = -\sqrt{4 \cdot 15} = -2\sqrt{15}

So, adding these together: 62+6−230−2156\sqrt{2} + 6 – 2\sqrt{30} – 2\sqrt{15}

Final Answer:

62−230+6−215\boxed{6\sqrt{2} – 2\sqrt{30} + 6 – 2\sqrt{15}}

This matches Option 1 in the image.

Explanation:

To find the product of two binomial expressions involving square roots, we treat them like any other binomials and use the distributive property, often referred to as FOIL (First, Outer, Inner, Last). In this problem, we simplify the square roots first: 12=23\sqrt{12} = 2\sqrt{3}. This reduces the expression to (23+6)(6−10)(2\sqrt{3} + \sqrt{6})(\sqrt{6} – \sqrt{10}).

We then apply FOIL:

  • First: Multiply 232\sqrt{3} by 6\sqrt{6} to get 2182\sqrt{18}, which simplifies to 626\sqrt{2}.
  • Outer: Multiply 232\sqrt{3} by −10-\sqrt{10} to get −230-2\sqrt{30}.
  • Inner: Multiply 6\sqrt{6} by 6\sqrt{6} to get 66.
  • Last: Multiply 6\sqrt{6} by −10-\sqrt{10} to get −60-\sqrt{60}, which simplifies to −215-2\sqrt{15}.

Adding all the terms, we get: 62+6−230−2156\sqrt{2} + 6 – 2\sqrt{30} – 2\sqrt{15}

This expression cannot be simplified further as all radical parts have different radicands. The order of terms may vary, but this matches exactly with the first multiple choice option. So, the correct answer is:

✅ 6√2 – 2√30 + 6 – 2√15.

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