If a=2b and b=4c, then `root3(a^(2)/(16bc))=` ____ .

Answer

To solve the problem step by step, we start with the given equations and work through the expression step by step.

### Step 1: Substitute the values of a and b in terms of c
We are given:
- \( a = 2b \)
- \( b = 4c \)

First, we can substitute \( b \) in the equation for \( a \):
\[ a = 2b = 2(4c) = 8c \]

### Step 2: Write the expression to be evaluated
We need to find:
\[ \sqrt[3]{\frac{a^2}{16bc}} \]

### Step 3: Substitute \( a \) and \( b \) in the expression
Now we substitute \( a = 8c \) and \( b = 4c \) into the expression:
\[ \sqrt[3]{\frac{(8c)^2}{16(4c)(c)}} \]

### Step 4: Simplify the expression
Calculate \( (8c)^2 \):
\[ (8c)^2 = 64c^2 \]
Now substitute this back into the expression:
\[ \sqrt[3]{\frac{64c^2}{16 \cdot 4c^2}} \]

Now calculate \( 16 \cdot 4c^2 \):
\[ 16 \cdot 4c^2 = 64c^2 \]
So the expression simplifies to:
\[ \sqrt[3]{\frac{64c^2}{64c^2}} = \sqrt[3]{1} \]

### Step 5: Calculate the cube root
The cube root of 1 is:
\[ \sqrt[3]{1} = 1 \]

### Final Answer
Thus, the value of \( \sqrt[3]{\frac{a^2}{16bc}} \) is:
\[ \boxed{1} \]