Math Problem Statement

solve

Simplify the expression \( \left( \frac{x^2}{y} \right)^{-3} \).

Solution

To simplify the expression (x2y)3\left( \frac{x^2}{y} \right)^{-3}, let's proceed step by step.

Given:

(x2y)3\left( \frac{x^2}{y} \right)^{-3}

Step 1: Apply the negative exponent rule

The rule states that ab=1aba^{-b} = \frac{1}{a^b}. Applying this: (x2y)3=1(x2y)3.\left( \frac{x^2}{y} \right)^{-3} = \frac{1}{\left( \frac{x^2}{y} \right)^3}.

Step 2: Expand the positive exponent

Raise the fraction x2y\frac{x^2}{y} to the power of 3: (x2y)3=(x2)3y3.\left( \frac{x^2}{y} \right)^3 = \frac{(x^2)^3}{y^3}.

Using the power rule (am)n=amn(a^m)^n = a^{m \cdot n}: (x2)3=x23=x6.(x^2)^3 = x^{2 \cdot 3} = x^6.

Thus: (x2y)3=x6y3.\left( \frac{x^2}{y} \right)^3 = \frac{x^6}{y^3}.

Step 3: Simplify the expression

Now substitute back: (x2y)3=1x6y3.\left( \frac{x^2}{y} \right)^{-3} = \frac{1}{\frac{x^6}{y^3}}.

Simplify the fraction: 1x6y3=y3x6.\frac{1}{\frac{x^6}{y^3}} = \frac{y^3}{x^6}.

Final Answer:

y3x6\boxed{\frac{y^3}{x^6}}

Let me know if you'd like more explanation!


Related Questions:

  1. What happens if the exponent were +3+3 instead of 3-3?
  2. How do you simplify an expression with multiple variables and exponents?
  3. What are the laws of exponents, and how are they applied to fractions?
  4. How does the negative exponent rule affect the structure of a fraction?
  5. Can you provide examples of real-world applications of exponent rules?

Tip:

When dealing with negative exponents, always remember to reciprocate the base and apply the positive power afterward!

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Math Problem Analysis

Mathematical Concepts

Exponents
Fractions
Algebra

Formulas

Negative exponent rule: \( a^{-b} = \frac{1}{a^b} \)
Power rule: \( (a^m)^n = a^{m \cdot n} \)

Theorems

Laws of Exponents

Suitable Grade Level

Grades 8-10