Please solve the problem: “2 plus the square root of 41 divided by the square root of 3 minus 2 multiplied by 40.
The Correct Answer and Explanation is:
Let’s break down and solve the problem step-by-step:
The given expression is: 2+413−2×402 + \frac{\sqrt{41}}{\sqrt{3}} – 2 \times 402+341−2×40
Step 1: Simplify 2×402 \times 402×40
First, calculate the multiplication: 2×40=802 \times 40 = 802×40=80
Now the expression becomes: 2+413−802 + \frac{\sqrt{41}}{\sqrt{3}} – 802+341−80
Step 2: Simplify the fraction 413\frac{\sqrt{41}}{\sqrt{3}}341
To simplify this fraction, we can rationalize the denominator. This involves multiplying both the numerator and the denominator by 3\sqrt{3}3, so: 413=41×33×3=1233\frac{\sqrt{41}}{\sqrt{3}} = \frac{\sqrt{41} \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{\sqrt{123}}{3}341=3×341×3=3123
Now the expression becomes: 2+1233−802 + \frac{\sqrt{123}}{3} – 802+3123−80
Step 3: Approximate 123\sqrt{123}123
We can approximate the square root of 123. Using a calculator: 123≈11.09\sqrt{123} \approx 11.09123≈11.09
Now substitute this into the expression: 2+11.093−802 + \frac{11.09}{3} – 802+311.09−80
Step 4: Divide 11.0911.0911.09 by 3
Now, calculate: 11.093≈3.70\frac{11.09}{3} \approx 3.70311.09≈3.70
Step 5: Final Calculation
Now substitute this value back into the expression: 2+3.70−802 + 3.70 – 802+3.70−80
Add and subtract: 2+3.70=5.702 + 3.70 = 5.702+3.70=5.70 5.70−80=−74.305.70 – 80 = -74.305.70−80=−74.30
Final Answer:
The value of the expression is approximately: −74.30\boxed{-74.30}−74.30
This is the simplified solution to the given problem.
