What is the following product? (5sqrt(2) – 4sqrt(3))(5sqrt(2) – 4sqrt(3))
-2
2
22 – 40sqrt(6)
98 – 40sqrt(6)
The Correct Answer and Explanation is :
The correct answer is: 98 – 40\sqrt{6}
Let’s simplify the expression step by step. The product you’re trying to find is:
[
(5\sqrt{2} – 4\sqrt{3})(5\sqrt{2} – 4\sqrt{3})
]
This is of the form ((a – b)(a – b)), which is a special case of the expansion formula:
[
(a – b)(a – b) = a^2 – 2ab + b^2
]
Here, (a = 5\sqrt{2}) and (b = 4\sqrt{3}). Applying the formula:
[
a^2 – 2ab + b^2
]
Now, let’s calculate each part:
- (a^2):
[
(5\sqrt{2})^2 = 5^2 \times (\sqrt{2})^2 = 25 \times 2 = 50
] - (b^2):
[
(4\sqrt{3})^2 = 4^2 \times (\sqrt{3})^2 = 16 \times 3 = 48
] - (-2ab):
[
-2 \times (5\sqrt{2}) \times (4\sqrt{3}) = -2 \times 5 \times 4 \times \sqrt{2} \times \sqrt{3} = -40 \times \sqrt{6}
]
Now, putting everything together:
[
a^2 – 2ab + b^2 = 50 – 40\sqrt{6} + 48
]
Simplify the constants:
[
50 + 48 = 98
]
So, the expression becomes:
[
98 – 40\sqrt{6}
]
Thus, the correct answer is:
[
\boxed{98 – 40\sqrt{6}}
]
Explanation:
This type of problem involves expanding binomials with square roots. The key is recognizing the pattern ((a – b)(a – b)), which can be expanded using the difference of squares formula (a^2 – 2ab + b^2). After calculating each term, we combine the constant parts and the radical parts separately. The result is a mixed number with both a constant term (98) and a radical term involving the square root of 6. This technique is common in algebra when simplifying products of square roots or binomials.