The rationalising factor of `root(5)(a^(2)b^(3)c^(4))` is

Options

A. `root(5)(a^(3)b^(2)c)`
B. `root(4)(a^(3)b^(2)c)`
C. `root(3)(a^(3)b^(2)c)`
D. `sqrt(a^(3)b^(2)c)`

Answer

To find the rationalizing factor of \( \sqrt[5]{a^2 b^3 c^4} \), we need to express it in a way that eliminates the root when multiplied by its rationalizing factor. Here’s a step-by-step solution:

### Step 1: Identify the expression
We start with the expression \( \sqrt[5]{a^2 b^3 c^4} \).

### Step 2: Rewrite the expression
We can rewrite the expression using fractional exponents:
\[ \sqrt[5]{a^2 b^3 c^4} = (a^2 b^3 c^4)^{1/5} \]

### Step 3: Determine the powers needed for rationalization
To rationalize, we need each base's exponent to be a multiple of 5. The current exponents are:
- For \( a \): 2
- For \( b \): 3
- For \( c \): 4

We need to find how much more we need to add to each exponent to make them multiples of 5:
- For \( a^2 \): We need \( 5 - 2 = 3 \) more, so we need \( a^3 \).
- For \( b^3 \): We need \( 5 - 3 = 2 \) more, so we need \( b^2 \).
- For \( c^4 \): We need \( 5 - 4 = 1 \) more, so we need \( c^1 \).

### Step 4: Write the rationalizing factor
Thus, the rationalizing factor is:
\[ a^3 b^2 c^1 = a^3 b^2 c \]

### Step 5: Final answer
The rationalizing factor of \( \sqrt[5]{a^2 b^3 c^4} \) is:
\[ \boxed{a^3 b^2 c} \]

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