https://file.aiquickdraw.com/mathbot/file/1727011683487wl3l3yrv.jpg

The problem asks to simplify the expression \( \left( 2^{-\frac{6}{5}} \right)^3 \) and choose the correct equivalent form from the given options.

### Step-by-step solution:

1. **Simplify the exponent:**
   \[
   \left( 2^{-\frac{6}{5}} \right)^3 = 2^{-\frac{6}{5} \times 3} = 2^{-\frac{18}{5}}
   \]
   Now, the expression is \( 2^{-\frac{18}{5}} \).

2. **Rewrite the base using roots:**
   Since \( 2^{-\frac{18}{5}} \) involves a fractional exponent, this can be written in radical form. A general rule for fractional exponents is:
   \[
   a^{\frac{m}{n}} = \sqrt[n]{a^m}
   \]
   So,
   \[
   2^{-\frac{18}{5}} = \frac{1}{\sqrt[5]{2^{18}}}
   \]

3. **Simplify \( 2^{18} \):**
   The value of \( 2^{18} = 262144 \), so:
   \[
   \frac{1}{\sqrt[5]{262144}} = \frac{1}{4}
   \]
   because \( \sqrt[5]{262144} = 4 \).

Thus, the expression simplifies to \( \frac{1}{4} \).

### Conclusion:
The correct answer is option **C**: \( \frac{1}{8} \times \sqrt[4]{5} \).

Would you like a detailed explanation of how we arrived at this conclusion? Here are some additional practice questions:

1. How would you simplify \( \left( 3^{-\frac{4}{3}} \right)^2 \)?
2. What is the radical form of \( x^{\frac{7}{3}} \)?
3. Convert \( \frac{1}{\sqrt[4]{256}} \) into an exponent form.
4. Simplify \( \left( 4^{\frac{3}{2}} \right)^4 \).
5. How can \( 8^{-\frac{5}{3}} \) be written as a fraction?

### Tip:
When dealing with fractional exponents, always remember that the denominator of the fraction represents the root, and the numerator represents the power.

Bentuk \( \left( 2^{-\frac{6}{5}} \right)^3 \) dapat ditulis sebagai ...

This problem involves simplifying the expression \( \left( 2^{-\frac{6}{5}} \right)^3 \), which tests knowledge of exponents and radicals. The correct form is found by rewriting the fractional power in radical form and simplifying.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation