To determine which of the following expressions matches the given radical expression [tex]\( 4 \sqrt[3]{8^2} \)[/tex], let's analyze each choice step by step.
Given radical expression:
[tex]\[ 4 \sqrt[3]{8^2} \][/tex]
### Step-by-Step Simplification:
#### Expression 1: [tex]\( 4 \sqrt[3]{8^2} \)[/tex]
First, simplify inside the cube root:
[tex]\[ 8^2 = 64 \][/tex]
So, the expression becomes:
[tex]\[ 4 \sqrt[3]{64} \][/tex]
We need to find the cube root of 64:
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
Thus,
[tex]\[ 4 \times 4 = 16 \][/tex]
This simplifies to 16.
#### Expression 2: [tex]\( 4 \sqrt[3]{8} \)[/tex]
Simplify inside the cube root:
[tex]\[ \sqrt[3]{8} = 2 \][/tex]
Thus,
[tex]\[ 4 \times 2 = 8 \][/tex]
This simplifies to 8.
#### Expression 3: [tex]\( \sqrt[5]{4 a^2} \)[/tex]
Assume a value for simplicity, say [tex]\( a = 1 \)[/tex]:
[tex]\[ \sqrt[5]{4 \cdot 1^2} = \sqrt[5]{4} \][/tex]
Since we are looking for a specific value and not matching in a detailed way, the simplification stops here. This expression is approximately
[tex]\[ \sqrt[5]{4} \][/tex] (simplified form, not a specific number we see immediately).
#### Expression 4: [tex]\( \sqrt[3]{48} \)[/tex]
Simplify the cube root:
[tex]\[ \sqrt[3]{48} \approx 3.634 \][/tex]
Approximation does not match exactly our specific requirements in terms of radical form simplifying to integer values evident in given options.
Thus, comparing all the simplified forms:
[tex]\[ 4 \sqrt[3]{8^2} = 16 \][/tex]
only Expression 1 gives us an exact match after simplification.
Therefore, the answer is:
[tex]\[ 4 \sqrt[3]{8^2} \][/tex] is the correct match, which corresponds to the first expression.
So, the radical expression that corresponds to [tex]\( 4 \sqrt[3]{8^2} \)[/tex] is:
[tex]\[ \boxed{1} \][/tex]
