To determine which expression is equivalent to [tex]\(\sqrt[3]{54} + \sqrt[3]{16}\)[/tex], we will compare the numeric approximations of each option with the given value.
First, let's list the options for easy reference:
(A) [tex]\(3 \sqrt[3]{6} + 4\)[/tex]
(B) [tex]\(3 \sqrt[3]{3} + 2 \sqrt[3]{2}\)[/tex]
(C) [tex]\(5\)[/tex]
(D) [tex]\(5 \sqrt[3]{2}\)[/tex]
(E) [tex]\(7\)[/tex]
Now, let's analyze each option to identify the one equivalent to [tex]\(\sqrt[3]{54} + \sqrt[3]{16}\)[/tex]:
1. [tex]\(\sqrt[3]{54}\)[/tex] and [tex]\(\sqrt[3]{16}\)[/tex] individually:
- [tex]\(\sqrt[3]{54} \approx 3.78\)[/tex]
- [tex]\(\sqrt[3]{16}\)[/tex] or [tex]\(2^{4/3} \approx 2.52\)[/tex]
Adding these together:
- [tex]\(\sqrt[3]{54} + \sqrt[3]{16} \approx 3.78 + 2.52 = 6.30\)[/tex]
Next, we approximate each choice to see which matches closely with [tex]\(6.30\)[/tex]:
A) [tex]\(3 \sqrt[3]{6} + 4\)[/tex]:
[tex]\[
3 \sqrt[3]{6} \approx 3 \times 1.82 = 5.46
\][/tex]
[tex]\[
3 \sqrt[3]{6} + 4 = 5.46 + 4 = 9.46
\][/tex]
B) [tex]\(3 \sqrt[3]{3} + 2 \sqrt[3]{2}\)[/tex]:
[tex]\[
3 \sqrt[3]{3} \approx 3 \times 1.44 = 4.32
\][/tex]
[tex]\[
2 \sqrt[3]{2} \approx 2 \times 1.26 = 2.52
\][/tex]
[tex]\[
3 \sqrt[3]{3} + 2 \sqrt[3]{2} = 4.32 + 2.52 = 6.84
\][/tex]
C) 5:
Directly, [tex]\(5\)[/tex] does not equal [tex]\(6.30\)[/tex].
D) [tex]\(5 \sqrt[3]{2}\)[/tex]:
[tex]\[
5 \sqrt[3]{2} \approx 5 \times 1.26 = 6.30
\][/tex]
E) 7:
Directly, [tex]\(7\)[/tex] does not equal [tex]\(6.30\)[/tex].
By comparing each result, we observe that:
- [tex]\(5 \sqrt[3]{2}\)[/tex] closely matches [tex]\(6.30\)[/tex].
Thus, the choice that is equivalent to [tex]\(\sqrt[3]{54} + \sqrt[3]{16}\)[/tex] is:
[tex]\[
\boxed{5 \sqrt[3]{2}}
\][/tex]
