The expression \(\left(\frac{a^{30}}{a^{10}}\right)^{-\frac{1}{3}}\) can be simplified as follows:

### Step 1: Simplify the fraction inside the parentheses.
Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify:

\[
\frac{a^{30}}{a^{10}} = a^{30-10} = a^{20}.
\]

### Step 2: Apply the outer exponent \(-\frac{1}{3}\).
Now we raise \(a^{20}\) to the power of \(-\frac{1}{3}\). Using the property \((a^m)^n = a^{m \cdot n}\), we get:

\[
\left(a^{20}\right)^{-\frac{1}{3}} = a^{20 \cdot -\frac{1}{3}} = a^{-\frac{20}{3}}.
\]

### Final Answer:
\[
a^{-\frac{20}{3}}.
\]

Would you like further details on any of these steps?

Here are 5 related questions:
1. How do you simplify expressions with negative exponents?
2. What happens when the base is raised to a fraction power?
3. How do we simplify expressions with multiple exponents?
4. How can you apply exponent rules when combining variables?
5. What happens when you take a reciprocal of an expression with a negative exponent?

**Tip:** Negative exponents indicate reciprocals. For instance, \(a^{-n} = \frac{1}{a^n}\).

Learn how to simplify the expression (a^30/a^10)^-1/3 step-by-step, including the application of exponent rules, simplifying fractions, and working with negative and fractional exponents. This problem is suitable for students in Grades 8-10 and covers core algebra concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation