If the positive square root of (root 90 + root 80 ) is multiplied by (root (2 -1)) and the product is raised to the power of four, the result would be
The Correct Answer and Explanation is:
To solve this problem, we will break it down step by step.
Step 1: Simplify the expression inside the square root
We start with:90+80\sqrt{90} + \sqrt{80}90+80
First, simplify each square root:90=9×10=310\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}90=9×10=31080=16×5=45\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}80=16×5=45
So, we now have:90+80=310+45\sqrt{90} + \sqrt{80} = 3\sqrt{10} + 4\sqrt{5}90+80=310+45
Step 2: Multiply the sum by 2−1\sqrt{2 – 1}2−1
Next, we are asked to multiply this sum by 2−1\sqrt{2 – 1}2−1. Since 2−1=12 – 1 = 12−1=1, we have:2−1=1=1\sqrt{2 – 1} = \sqrt{1} = 12−1=1=1
Thus, multiplying by 1 does not change the value, so:(310+45)×1=310+45(3\sqrt{10} + 4\sqrt{5}) \times 1 = 3\sqrt{10} + 4\sqrt{5}(310+45)×1=310+45
Step 3: Raise the product to the power of 4
Now, the product is raised to the power of 4. We need to simplify the entire expression 310+453\sqrt{10} + 4\sqrt{5}310+45 raised to the power of 4. However, we must note that:(310+45)4(3\sqrt{10} + 4\sqrt{5})^4(310+45)4
This is a bit more complex. Rather than expanding manually, let’s estimate each square root value for a rough result:10≈3.162\sqrt{10} \approx 3.16210≈3.1625≈2.236\sqrt{5} \approx 2.2365≈2.236
So,310≈3×3.162=9.4863\sqrt{10} \approx 3 \times 3.162 = 9.486310≈3×3.162=9.48645≈4×2.236=8.9444\sqrt{5} \approx 4 \times 2.236 = 8.94445≈4×2.236=8.944
Thus: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
Now, raise this to the power of 4:(18.43)4≈18443.62≈339,141,304(18.43)^4 \approx 18443.6^2 \approx 339,141,304(18.43)4≈18443.62≈339,141,304
Final Answer:
The result of raising the expression to the power of 4 is approximately 339,141,304.
