Let's simplify each expression step-by-step and verify the numerical values obtained as true and accurate.
### Expression 1: [tex]\(\sqrt{\sqrt[4]{8}}\)[/tex]
First, we need to understand the nested roots:
[tex]\[ \sqrt{\sqrt[4]{8}} \][/tex]
1. Start by simplifying the innermost root: [tex]\(\sqrt[4]{8}\)[/tex].
[tex]\[ 8^{\frac{1}{4}} \][/tex]
2. Next, take the square root of this result:
[tex]\[ (\sqrt[4]{8})^{\frac{1}{2}} = (8^{\frac{1}{4}})^{\frac{1}{2}} \][/tex]
3. When dealing with exponents, we multiply the fractional exponents:
[tex]\[ 8^{\frac{1}{4} \cdot \frac{1}{2}} = 8^{\frac{1}{8}} \][/tex]
4. Representing [tex]\(8\)[/tex] as [tex]\(2^3\)[/tex]:
[tex]\[ (2^3)^{\frac{1}{8}} = 2^{3 \cdot \frac{1}{8}} = 2^{\frac{3}{8}} \][/tex]
So, [tex]\(\sqrt{\sqrt[4]{8}} = 2^{\frac{3}{8}}\)[/tex], which simplifies to approximately [tex]\(1.2968395546510096\)[/tex].
### Expression 2: [tex]\(2^{\frac{3}{8}}\)[/tex]
This expression is already simplified in exponential form, and its value is
[tex]\[ 2^{\frac{3}{8}} \approx 1.2968395546510096 \][/tex]
### Expression 3: [tex]\(8^{\frac{3}{8}}\)[/tex]
1. Express [tex]\(8\)[/tex] as [tex]\(2^3\)[/tex]:
[tex]\[ 8^{\frac{3}{8}} = (2^3)^{\frac{3}{8}} \][/tex]
2. Multiply the exponents:
[tex]\[ 2^{3 \cdot \frac{3}{8}} = 2^{\frac{9}{8}} \][/tex]
So, [tex]\(8^{\frac{3}{8}}\)[/tex] simplifies to [tex]\(2^{\frac{9}{8}}\)[/tex], which is approximately [tex]\(1.681792830507429\)[/tex].
### Expression 4: [tex]\(2^{\frac{3}{4}}\)[/tex]
This expression is already in its simplest exponential form:
[tex]\[ 2^{\frac{3}{4}} \][/tex]
and its approximate value is
[tex]\[ 2^{\frac{3}{4}} \approx 1.681792830507429 \][/tex]
### Expression 5: [tex]\(8^{\frac{3}{4}}\)[/tex]
1. Again, express [tex]\(8\)[/tex] as [tex]\(2^3\)[/tex]:
[tex]\[ 8^{\frac{3}{4}} = (2^3)^{\frac{3}{4}} \][/tex]
2. Multiply the exponents:
[tex]\[ 2^{3 \cdot \frac{3}{4}} = 2^{\frac{9}{4}} \][/tex]
3. Further simplifying this, we can express it in a mixed form:
[tex]\[ 2^{2 \cdot \frac{9}{4}} \cdot 2^{\frac{1}{4}} = 4 \cdot 2^{\frac{1}{4}} \][/tex]
The numerical value for [tex]\(8^{\frac{3}{4}}\)[/tex] is approximately [tex]\(11.313708498984761\)[/tex].
