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

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we start with the given equations and substitute accordingly.

### Step 1: Understand the relationships
We are given:
- \( a = 2b \)
- \( b = 4c \)

### Step 2: Substitute \( b \) in terms of \( c \) into \( a \)
From the second equation, we can substitute \( b \) into the first equation:
\[ b = 4c \implies a = 2(4c) = 8c \]
So, we have:
\[ a = 8c \]

### Step 3: Substitute \( a \) and \( b \) into the expression
We need to evaluate:
\[ \sqrt[3]{\frac{a^2}{16bc}} \]
Substituting \( a \) and \( b \):
\[ \sqrt[3]{\frac{(8c)^2}{16(4c)(c)}} \]

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

### Step 5: Simplify the denominator
Calculating the denominator:
\[ 16(4c^2) = 64c^2 \]
Now the expression becomes:
\[ \sqrt[3]{\frac{64c^2}{64c^2}} = \sqrt[3]{1} \]

### Step 6: 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:
\[ \boxed{1} \]

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