https://r2.erweima.ai/mathbot/image/1726413952099wddfxxb9.jpg

Let's break down and simplify the given expression step by step. The original expression is:

\[
\left( \frac{3^{-\frac{1}{6}} X^{-\frac{1}{3}}}{y^6 m^3} \right)^3 \times \frac{(X^{-1} m^{-2})^{-\frac{1}{2}}}{2 y^{-2}}
\]

### Step 1: Simplify the powers inside the parentheses

First, simplify the first part of the expression:

\[
\left( \frac{3^{-\frac{1}{6}} X^{-\frac{1}{3}}}{y^6 m^3} \right)^3
\]

Using the rule of exponents \((a^m / b^n)^p = a^{mp} / b^{np}\):

\[
= \frac{(3^{-\frac{1}{6}})^3 (X^{-\frac{1}{3}})^3}{(y^6)^3 (m^3)^3}
= \frac{3^{-\frac{1}{2}} X^{-1}}{y^{18} m^9}
\]

### Step 2: Simplify the second part of the expression

Now, simplify the second part:

\[
(X^{-1} m^{-2})^{-\frac{1}{2}} = X^{\frac{1}{2}} m^1
\]

This follows from the exponent rule \((a^m)^n = a^{m \cdot n}\).

### Step 3: Multiply both parts together

Now, multiply the two simplified expressions:

\[
\frac{3^{-\frac{1}{2}} X^{-1}}{y^{18} m^9} \times \frac{X^{\frac{1}{2}} m^1}{2 y^{-2}}
\]

First, simplify the \( y \) terms:

\[
= \frac{3^{-\frac{1}{2}} X^{-1} X^{\frac{1}{2}} m^1}{2 y^{18} y^{-2} m^9}
= \frac{3^{-\frac{1}{2}} X^{-\frac{1}{2}} m}{2 y^{20} m^9}
\]

### Step 4: Final simplification

Now, simplify the \( m \) and \( X \) terms:

\[
= \frac{3^{-\frac{1}{2}} m^{-8} X^{-\frac{1}{2}}}{2 y^{20}}
\]

This is the fully simplified expression:

\[
\frac{3^{-\frac{1}{2}} X^{-\frac{1}{2}} m^{-8}}{2 y^{20}}
\]

Would you like to dive deeper into any specific part of this solution?

### Follow-up questions:
1. How do we apply the power rule when raising a fraction to a power?
2. Can you clarify how exponents of negative powers are simplified?
3. How do you handle variables with different exponents when multiplying them?
4. What are the general rules for handling square roots in terms of exponents?
5. How would this expression change if the exponents inside the parentheses were positive?

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

Simplify the following expression: \[ \left( \frac{3^{-\frac{1}{6}} X^{-\frac{1}{3}}}{y^6 m^3} \right)^3 \times \frac{(X^{-1} m^{-2})^{-\frac{1}{2}}}{2 y^{-2}} \]

Learn how to simplify complex exponent expressions like \left( \frac{3^{-\frac{1}{6}} X^{-\frac{1}{3}}}{y^6 m^3} \right)^3 \times \frac{(X^{-1} m^{-2})^{-\frac{1}{2}}}{2 y^{-2}} using exponent rules. This solution involves the step-by-step application of algebraic concepts, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation