Answer :
Certainly! Let's tackle this problem step by step.
The expression to simplify is:
[tex]\[ \sqrt{\sqrt{25} + \sqrt{121}} - \sqrt[3]{2 - \sqrt[3]{3 \sqrt{81}}} + \sqrt[5]{\sqrt{-8} \cdot \sqrt{\sqrt{16}} \cdot \sqrt{64}} \][/tex]
### Step 1: Simplify [tex]\(\sqrt{\sqrt{25} + \sqrt{121}}\)[/tex]
1. Compute [tex]\(\sqrt{25}\)[/tex]:
[tex]\[ \sqrt{25} = 5 \][/tex]
2. Compute [tex]\(\sqrt{121}\)[/tex]:
[tex]\[ \sqrt{121} = 11 \][/tex]
3. Add the results:
[tex]\[ \sqrt{25} + \sqrt{121} = 5 + 11 = 16 \][/tex]
4. Compute the square root of the sum:
[tex]\[ \sqrt{16} = 4 \][/tex]
Thus, the first part simplifies to 4.
### Step 2: Simplify [tex]\(\sqrt[3]{2 - \sqrt[3]{3 \sqrt{81}}}\)[/tex]
1. Compute [tex]\(\sqrt{81}\)[/tex]:
[tex]\[ \sqrt{81} = 9 \][/tex]
2. Multiply by 3:
[tex]\[ 3 \cdot \sqrt{81} = 3 \cdot 9 = 27 \][/tex]
3. Compute the cube root of 27:
[tex]\[ \sqrt[3]{27} = 3 \][/tex]
4. Subtract this result from 2:
[tex]\[ 2 - \sqrt[3]{27} = 2 - 3 = -1 \][/tex]
5. Compute the cube root of -1:
[tex]\[ \sqrt[3]{-1} = -1 \][/tex]
Thus, the second part simplifies to -1.
### Step 3: Simplify [tex]\(\sqrt[5]{\sqrt{-8} \cdot \sqrt{\sqrt{16}} \cdot \sqrt{64}}\)[/tex]
1. Compute [tex]\(\sqrt{-8}\)[/tex]:
Since [tex]\(-8\)[/tex] is negative, we'll move into complex numbers for this part:
[tex]\[ \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2} \cdot i \approx 2.8284271247461903 \cdot i \][/tex]
2. Compute [tex]\(\sqrt{\sqrt{16}}\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
[tex]\[ \sqrt{4} = 2 \][/tex]
3. Compute [tex]\(\sqrt{64}\)[/tex]:
[tex]\[ \sqrt{64} = 8 \][/tex]
4. Multiply all results:
[tex]\[ 2\sqrt{2} \cdot i \cdot 2 \cdot 8 = 2 \cdot 2 \cdot 8 \cdot \sqrt{2} \cdot i = 32 \sqrt{2} \cdot i \][/tex]
5. Compute the fifth root of the product:
[tex]\[ \sqrt[5]{32 \sqrt{2} \cdot i} \][/tex]
Breaking it down, we express [tex]\(32 \sqrt{2}\)[/tex] as:
[tex]\[ 32 \sqrt{2} = (2^5 \cdot 2^{1/2}) = 2^{5.5} \][/tex]
Thus:
[tex]\[ \sqrt[5]{2^{5.5} \cdot i} = 2^{5.5/5} \cdot \sqrt[5]{i} = 2^{1.1} \cdot \sqrt[5]{i} \][/tex]
Approximate value:
[tex]\[ 2^{1.1} \approx 2.1435469 \][/tex]
Thus, the third part simplifies to approximately [tex]\(2.1435469 \cdot \sqrt[5]{i}\)[/tex].
### Final Step:
Sum the parts obtained:
[tex]\[ 4 - (-1) + (\approx 2.1435469 \cdot \sqrt[5]{i}) \][/tex]
Thus the final answer is the sum of the real parts and the imaginary parts respectively:
[tex]\[ \approx 5 + 2.1435469 \cdot \sqrt[5]{i} \][/tex]
However, for substantive calculations, you may need to deal with the result close enough accurate up to complex fifth roots.
Finally, approximations leave the resultant answer closer to:
[tex]\[ 4 - (-1) \text{(real part approximation)} = 5 + \text{complex contribution} \][/tex]
Detailed breakdown indeed involves thinking through accurate fifth roots with complex jogs mainly. Feel free to pursue numerics for deeper analysis.
The expression to simplify is:
[tex]\[ \sqrt{\sqrt{25} + \sqrt{121}} - \sqrt[3]{2 - \sqrt[3]{3 \sqrt{81}}} + \sqrt[5]{\sqrt{-8} \cdot \sqrt{\sqrt{16}} \cdot \sqrt{64}} \][/tex]
### Step 1: Simplify [tex]\(\sqrt{\sqrt{25} + \sqrt{121}}\)[/tex]
1. Compute [tex]\(\sqrt{25}\)[/tex]:
[tex]\[ \sqrt{25} = 5 \][/tex]
2. Compute [tex]\(\sqrt{121}\)[/tex]:
[tex]\[ \sqrt{121} = 11 \][/tex]
3. Add the results:
[tex]\[ \sqrt{25} + \sqrt{121} = 5 + 11 = 16 \][/tex]
4. Compute the square root of the sum:
[tex]\[ \sqrt{16} = 4 \][/tex]
Thus, the first part simplifies to 4.
### Step 2: Simplify [tex]\(\sqrt[3]{2 - \sqrt[3]{3 \sqrt{81}}}\)[/tex]
1. Compute [tex]\(\sqrt{81}\)[/tex]:
[tex]\[ \sqrt{81} = 9 \][/tex]
2. Multiply by 3:
[tex]\[ 3 \cdot \sqrt{81} = 3 \cdot 9 = 27 \][/tex]
3. Compute the cube root of 27:
[tex]\[ \sqrt[3]{27} = 3 \][/tex]
4. Subtract this result from 2:
[tex]\[ 2 - \sqrt[3]{27} = 2 - 3 = -1 \][/tex]
5. Compute the cube root of -1:
[tex]\[ \sqrt[3]{-1} = -1 \][/tex]
Thus, the second part simplifies to -1.
### Step 3: Simplify [tex]\(\sqrt[5]{\sqrt{-8} \cdot \sqrt{\sqrt{16}} \cdot \sqrt{64}}\)[/tex]
1. Compute [tex]\(\sqrt{-8}\)[/tex]:
Since [tex]\(-8\)[/tex] is negative, we'll move into complex numbers for this part:
[tex]\[ \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2} \cdot i \approx 2.8284271247461903 \cdot i \][/tex]
2. Compute [tex]\(\sqrt{\sqrt{16}}\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
[tex]\[ \sqrt{4} = 2 \][/tex]
3. Compute [tex]\(\sqrt{64}\)[/tex]:
[tex]\[ \sqrt{64} = 8 \][/tex]
4. Multiply all results:
[tex]\[ 2\sqrt{2} \cdot i \cdot 2 \cdot 8 = 2 \cdot 2 \cdot 8 \cdot \sqrt{2} \cdot i = 32 \sqrt{2} \cdot i \][/tex]
5. Compute the fifth root of the product:
[tex]\[ \sqrt[5]{32 \sqrt{2} \cdot i} \][/tex]
Breaking it down, we express [tex]\(32 \sqrt{2}\)[/tex] as:
[tex]\[ 32 \sqrt{2} = (2^5 \cdot 2^{1/2}) = 2^{5.5} \][/tex]
Thus:
[tex]\[ \sqrt[5]{2^{5.5} \cdot i} = 2^{5.5/5} \cdot \sqrt[5]{i} = 2^{1.1} \cdot \sqrt[5]{i} \][/tex]
Approximate value:
[tex]\[ 2^{1.1} \approx 2.1435469 \][/tex]
Thus, the third part simplifies to approximately [tex]\(2.1435469 \cdot \sqrt[5]{i}\)[/tex].
### Final Step:
Sum the parts obtained:
[tex]\[ 4 - (-1) + (\approx 2.1435469 \cdot \sqrt[5]{i}) \][/tex]
Thus the final answer is the sum of the real parts and the imaginary parts respectively:
[tex]\[ \approx 5 + 2.1435469 \cdot \sqrt[5]{i} \][/tex]
However, for substantive calculations, you may need to deal with the result close enough accurate up to complex fifth roots.
Finally, approximations leave the resultant answer closer to:
[tex]\[ 4 - (-1) \text{(real part approximation)} = 5 + \text{complex contribution} \][/tex]
Detailed breakdown indeed involves thinking through accurate fifth roots with complex jogs mainly. Feel free to pursue numerics for deeper analysis.
