Which of the following is equivalent to [tex]\sqrt[3]{54} + \sqrt[3]{16}[/tex]?

A. [tex]3 \sqrt[3]{6} + 4[/tex]
B. [tex]3 \sqrt[3]{3} + 2 \sqrt[3]{2}[/tex]
C. 5
D. [tex]5 \sqrt[3]{2}[/tex]
E. 7



Answer :

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]