Let's evaluate the function [tex]\( S(x) = \sqrt[4]{8x^3} \)[/tex] at two specific values, [tex]\( x = 0 \)[/tex] and [tex]\( x = 8 \)[/tex].
### Part (a) [tex]\( S(0) \)[/tex]
To find [tex]\( S(0) \)[/tex]:
1. Substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( S(x) = \sqrt[4]{8x^3} \)[/tex]:
[tex]\( S(0) = \sqrt[4]{8 \cdot 0^3} \)[/tex]
2. Simplify inside the fourth root:
[tex]\( 0^3 = 0 \)[/tex]
3. Multiply by the constant 8:
[tex]\( 8 \cdot 0 = 0 \)[/tex]
4. Take the fourth root of 0:
[tex]\( \sqrt[4]{0} = 0 \)[/tex]
Therefore, [tex]\( S(0) = 0.0 \)[/tex].
### Part (b) [tex]\( S(8) \)[/tex]
To find [tex]\( S(8) \)[/tex]:
1. Substitute [tex]\( x = 8 \)[/tex] into the function [tex]\( S(x) = \sqrt[4]{8x^3} \)[/tex]:
[tex]\( S(8) = \sqrt[4]{8 \cdot 8^3} \)[/tex]
2. Calculate [tex]\( 8^3 \)[/tex]:
[tex]\( 8^3 = 512 \)[/tex]
3. Multiply by the constant 8:
[tex]\( 8 \cdot 512 = 4096 \)[/tex]
4. Take the fourth root of 4096:
[tex]\( \sqrt[4]{4096} = 8 \)[/tex]
Therefore, [tex]\( S(8) = 8.0 \)[/tex].
Summarizing the results:
a) [tex]\( S(0) = 0.0 \)[/tex]
b) [tex]\( S(8) = 8.0 \)[/tex]
