https://file.aiquickdraw.com/mathbot/file/1726381828229ez8f74xe.jpg

The problem in the image involves evaluating a product of two expressions with fractional exponents:

\[
\left( \frac{1}{81} \right)^{\frac{1}{2}} \cdot \left( \frac{1}{81} \right)^{\frac{3}{4}}
\]

### Steps to Solve:

1. **Combine the exponents**: When multiplying powers with the same base, we add the exponents. The base here is \(\frac{1}{81}\).
   \[
   \left( \frac{1}{81} \right)^{\frac{1}{2} + \frac{3}{4}} = \left( \frac{1}{81} \right)^{\frac{1}{2} + \frac{3}{4}}
   \]

2. **Add the exponents**:
   \[
   \frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}
   \]
   So, we now have:
   \[
   \left( \frac{1}{81} \right)^{\frac{5}{4}}
   \]

3. **Evaluate the expression**:
   \[
   \left( \frac{1}{81} \right)^{\frac{5}{4}} = \frac{1^{\frac{5}{4}}}{81^{\frac{5}{4}}}
   \]
   Since \(1^{\frac{5}{4}} = 1\), we focus on \(81^{\frac{5}{4}}\).

4. **Simplify \(81^{\frac{5}{4}}\)**:
   \[
   81 = 3^4
   \]
   Therefore:
   \[
   81^{\frac{5}{4}} = \left(3^4\right)^{\frac{5}{4}} = 3^5 = 243
   \]

5. **Final answer**:
   \[
   \frac{1}{243}
   \]

Thus, the value of the given expression is \(\frac{1}{243}\).

Would you like further details on this process? Here are some related questions to consider:

1. What happens when you multiply two terms with the same base but different exponents?
2. How do you simplify expressions with fractional exponents?
3. Can you explain how to add fractions with different denominators?
4. What is the general rule for exponentiation with powers of fractions?
5. How would the result change if we had different powers, say \(\frac{1}{3}\) and \(\frac{5}{6}\)?

**Tip**: When working with fractional exponents, remember that the denominator represents the root and the numerator represents the power.

Evaluate the expression (1/81)^(1/2) * (1/81)^(3/4).

Learn how to evaluate the expression (1/81)^(1/2) * (1/81)^(3/4) using the properties of exponents. This step-by-step guide simplifies the calculation and demonstrates how to handle fractional exponents. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation