What is the value of ?
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Squaring the difference gives 18 minus twice the product. The product under the root simplifies to 4, so the answer is 10. I'm pretty sure it's A.
Yes, I did the same. The product is sqrt(81-65) = sqrt(16) = 4, so 18 - 8 = 10.
That's clever. I initially tried to simplify each radical separately and got stuck. Squaring first is definitely the way to go.
Is there a way to solve this without squaring? I tried rationalizing but it got messy.
Not really. Squaring is the standard approach for expressions like this. The difference of squares under the radical is the key.
This one was tricky. I got confused with the signs. After squaring, you have to remember that the cross term is negative, so it's 18 minus 2 times the product. The product is 4, so 18 - 8 = 10. Answer A.
What it tests
Your handling of square and higher roots, including simplifying radicals.
Common trap
Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.