https://r2.erweima.ai/mathbot/image/1725939655084hiurjcsm.jpg

Let's interpret the mathematical expressions in the image step by step.

The image contains two separate expressions.

### First Expression (on the left):
\[
\frac{\sqrt[5]{2^3 \cdot x^7}}{\sqrt[5]{y^6}}
\]

### Second Expression (on the right):
\[
\frac{\sqrt[5]{y^{-1}}}{\sqrt[5]{y^{-1}}}
\]

#### Let's simplify each expression.

### First Expression Simplification:
1. Apply the root rule: \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \).
2. The expression becomes:
   \[
   \frac{(2^3 \cdot x^7)^{\frac{1}{5}}}{y^{\frac{6}{5}}}
   \]
3. Distribute the exponent \(\frac{1}{5}\) inside the numerator:
   \[
   \frac{2^{\frac{3}{5}} \cdot x^{\frac{7}{5}}}{y^{\frac{6}{5}}}
   \]

### Second Expression Simplification:
1. Both the numerator and denominator are identical: 
   \[
   \frac{\sqrt[5]{y^{-1}}}{\sqrt[5]{y^{-1}}} = 1
   \]
   Since any non-zero value divided by itself is equal to 1.

### Final Simplified Forms:
- First expression: \(\frac{2^{\frac{3}{5}} \cdot x^{\frac{7}{5}}}{y^{\frac{6}{5}}}\)
- Second expression: \(1\)

If you have any questions or need further details, feel free to ask!

### Related Questions:
1. What is the general rule for simplifying expressions with roots and exponents?
2. How do you combine terms with the same base but different exponents?
3. What happens when you raise a product of variables to a power?
4. How do you simplify expressions where both numerator and denominator are the same?
5. How does the rule \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \) work for negative exponents?

### Tip:
When simplifying roots and exponents, remember to always break down complex expressions into smaller parts and apply the basic exponentiation rules step by step.

The user submitted an image containing two mathematical expressions involving roots and exponents.

This page explains how to simplify mathematical expressions involving exponents and roots. The example problem covers exponentiation and radical rules, showing step-by-step how to handle fractions and simplify terms. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation