Let's determine the value of each expression in the list and match them with the given whole numbers.
1. [tex]\( 1,296^{\frac{3}{4}} \)[/tex]
To evaluate [tex]\( 1,296^{\frac{3}{4}} \)[/tex], we can consider the fourth root of 1296 and then raising the result to the power of 3.
[tex]\[
1,296^{\frac{3}{4}} = 216
\][/tex]
2. [tex]\( \left(6,561^{\frac{1}{4}}\right)^3 \)[/tex]
First, take the fourth root of 6,561, then raise the result to the power of 3.
[tex]\[
\left(6,561^{\frac{1}{4}}\right)^3 = 729
\][/tex]
3. [tex]\( \sqrt[4]{81^3} \)[/tex]
First, compute [tex]\( 81^3 \)[/tex], then take the fourth root of the result.
[tex]\[
\sqrt[4]{81^3} = 27
\][/tex]
4. [tex]\( 16^{\frac{3}{4}} \)[/tex]
Evaluate the fourth root of 16, then raise the result to the power of 3.
[tex]\[
16^{\frac{3}{4}} = 8
\][/tex]
5. [tex]\( \left(8^4\right)^{\frac{3}{4}} \)[/tex]
Raise 8 to the fourth power, then take that result and raise it to the power of [tex]\( \frac{3}{4} \)[/tex].
[tex]\[
\left(8^4\right)^{\frac{3}{4}} = 512
\][/tex]
6. [tex]\( (\sqrt[4]{2,401})^3 \)[/tex]
Take the fourth root of 2,401, then raise the result to the power of 3.
[tex]\[
(\sqrt[4]{2,401})^3 = 343
\][/tex]
Now let's match these calculated values with the given whole numbers:
- [tex]\( 1,296^{\frac{3}{4}} = 216 \)[/tex]
- [tex]\( \left(6,561^{\frac{1}{4}}\right)^3 = 729 \)[/tex]
- [tex]\( \sqrt[4]{81^3} = 27 \)[/tex]
- [tex]\( 16^{\frac{3}{4}} = 8 \)[/tex]
- [tex]\( \left(8^4\right)^{\frac{3}{4}} = 512 \)[/tex]
- [tex]\( (\sqrt[4]{2,401})^3 = 343 \)[/tex]
Matching each expression to the given whole numbers:
1. [tex]\( 1,296^{\frac{3}{4}} \)[/tex] → 216
2. [tex]\( \left(6,561^{\frac{1}{4}}\right)^3 \)[/tex] → 729
3. [tex]\( \sqrt[4]{81^3} \)[/tex] → 27
4. [tex]\( 16^{\frac{3}{4}} \)[/tex] → 8
5. [tex]\( \left(8^4\right)^{\frac{3}{4}} \)[/tex] → 512
6. [tex]\( (\sqrt[4]{2,401})^3 \)[/tex] → 343
