If the positive square root of (sqrt(90) + sqrt(80)) is multiplied by (sqrt(2) – 1) and the product is raised to the power of four, the result would be: 1. 100 2. 1600 3. 11520000 4. 10
The Correct Answer and Explanation is:
We are given the expression:
(90+80⋅(2−1))4(\sqrt{\sqrt{90} + \sqrt{80}} \cdot (\sqrt{2} – 1))^4(90+80⋅(2−1))4
We will simplify this step by step.
Step 1: Simplify the inner square roots
We begin with:90+80\sqrt{90} + \sqrt{80}90+80
Break these into their simplified radical form:
- 90=9⋅10=310\sqrt{90} = \sqrt{9 \cdot 10} = 3\sqrt{10}90=9⋅10=310
- 80=16⋅5=45\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}80=16⋅5=45
So we get:90+80=310+45\sqrt{90} + \sqrt{80} = 3\sqrt{10} + 4\sqrt{5}90+80=310+45
Step 2: Take the square root of the sum
We now compute:310+45\sqrt{3\sqrt{10} + 4\sqrt{5}}310+45
This is complicated to simplify exactly, so let’s approximate using decimals:
- 10≈3.162\sqrt{10} \approx 3.16210≈3.162, so 310≈3⋅3.162=9.4863\sqrt{10} \approx 3 \cdot 3.162 = 9.486310≈3⋅3.162=9.486
- 5≈2.236\sqrt{5} \approx 2.2365≈2.236, so 45≈4⋅2.236=8.9444\sqrt{5} \approx 4 \cdot 2.236 = 8.94445≈4⋅2.236=8.944
Now add:310+45≈9.486+8.944=18.433\sqrt{10} + 4\sqrt{5} \approx 9.486 + 8.944 = 18.43310+45≈9.486+8.944=18.43
Then take the square root:18.43≈4.29\sqrt{18.43} \approx 4.2918.43≈4.29
Step 3: Multiply by 2−1\sqrt{2} – 12−1
2≈1.414\sqrt{2} \approx 1.4142≈1.414, so 2−1≈0.414\sqrt{2} – 1 \approx 0.4142−1≈0.414
Now multiply:4.29⋅0.414≈1.7764.29 \cdot 0.414 \approx 1.7764.29⋅0.414≈1.776
Step 4: Raise the result to the power of 4
(1.776)4≈(1.7762)2=(3.154)2≈9.95(1.776)^4 \approx (1.776^2)^2 = (3.154)^2 \approx 9.95(1.776)4≈(1.7762)2=(3.154)2≈9.95
Rounded to the nearest whole number, the answer is:10\boxed{10}10
Final Answer:
Option 4: 10
Explanation:
The problem requires simplifying a compound expression involving nested radicals and exponentiation. First, we simplify the square roots of 90 and 80 using their prime factors. These simplify to 3103\sqrt{10}310 and 454\sqrt{5}45 respectively. Their sum does not simplify further using exact radicals, so we approximate to a decimal. Taking the square root of this sum, we get an estimate around 4.29. Multiplying by 2−1\sqrt{2} – 12−1 (approximately 0.414) gives roughly 1.776. Finally, raising this value to the 4th power gives close to 10. This shows how estimation using known square root approximations can help evaluate seemingly complicated expressions.
