The index form of `root9(((4)/(5))^(2))` is

Options

A. `((4)/(5))^(2/9)`
B. `((4)/(5))^(3)`
C. `((4)/(5))^(1//2)`
D. `((4)/(5))^(1//27)`

Answer

To find the index form of \( \sqrt[9]{\left(\frac{4}{5}\right)^2} \), we will follow these steps:

### Step 1: Rewrite the expression using exponent notation
The expression \( \sqrt[9]{\left(\frac{4}{5}\right)^2} \) can be rewritten using exponent notation. The \( n \)-th root of a number can be expressed as that number raised to the power of \( \frac{1}{n} \).

\[ \sqrt[9]{\left(\frac{4}{5}\right)^2} = \left(\left(\frac{4}{5}\right)^2\right)^{\frac{1}{9}} \]

### Step 2: Apply the power of a power property
Next, we apply the power of a power property, which states that \( (a^m)^n = a^{m \cdot n} \). In our case, we have:

\[ \left(\left(\frac{4}{5}\right)^2\right)^{\frac{1}{9}} = \left(\frac{4}{5}\right)^{2 \cdot \frac{1}{9}} = \left(\frac{4}{5}\right)^{\frac{2}{9}} \]

### Step 3: Express the fraction in terms of its base components
Now we can express \( \left(\frac{4}{5}\right)^{\frac{2}{9}} \) in terms of its base components:

\[ \left(\frac{4}{5}\right)^{\frac{2}{9}} = \frac{4^{\frac{2}{9}}}{5^{\frac{2}{9}}} \]

### Step 4: Write the final index form
Thus, the index form of \( \sqrt[9]{\left(\frac{4}{5}\right)^2} \) is:

\[ \frac{4^{\frac{2}{9}}}{5^{\frac{2}{9}}} \]

### Final Answer
The index form of \( \sqrt[9]{\left(\frac{4}{5}\right)^2} \) is \( \frac{4^{\frac{2}{9}}}{5^{\frac{2}{9}}} \).

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