Math Problem Statement

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

Solution

The expression in the image is:

(x2y2)3×(x2y3)\left( \frac{x^2}{y^{-2}} \right)^{-3} \times \left( \frac{x^2}{y^3} \right)

Let's simplify it step-by-step.

Step 1: Simplify each term individually

  1. First Term: (x2y2)3\left( \frac{x^2}{y^{-2}} \right)^{-3}

    When raising a fraction to a power, each component of the fraction is raised to that power. Therefore: (x2y2)3=(x2)3(y2)3=x6y6=1x6y6\left( \frac{x^2}{y^{-2}} \right)^{-3} = \frac{(x^2)^{-3}}{(y^{-2})^{-3}} = \frac{x^{-6}}{y^6} = \frac{1}{x^6 y^6}

  2. Second Term: x2y3\frac{x^2}{y^3} (this one doesn't need any additional simplification)

Step 2: Multiply the Simplified Terms

Now, we have: 1x6y6×x2y3=x2x6y6y3\frac{1}{x^6 y^6} \times \frac{x^2}{y^3} = \frac{x^2}{x^6 y^6 y^3}

Combine like terms in the denominator:

  • For xx: x2x^2 in the numerator and x6x^6 in the denominator combine as x26=x4x^{2 - 6} = x^{-4}.
  • For yy: y6y^6 and y3y^3 in the denominator combine as y6+3=y9y^{6 + 3} = y^9.

Thus, the expression becomes: x2x6y6y3=1x4y9=x4y9\frac{x^2}{x^6 y^6 y^3} = \frac{1}{x^4 y^9} = x^{-4} y^{-9}

Final Answer

1x4y9 or x4y9\boxed{\frac{1}{x^4 y^9} \text{ or } x^{-4} y^{-9}}

Would you like further details or have any questions about the steps?


Related Questions

  1. How would you simplify (x3y1)2×(x4y2)\left( \frac{x^3}{y^{-1}} \right)^{-2} \times \left( \frac{x^4}{y^2} \right)?
  2. How does multiplying two expressions with exponents affect the exponents?
  3. What happens when you raise a fraction with negative exponents to another negative exponent?
  4. How can you rewrite an expression with negative exponents as a fraction with only positive exponents?
  5. How would you handle an expression like (x3y2)4\left( \frac{x^{-3}}{y^2} \right)^4?

Tip

Remember, raising an expression with an exponent in the denominator to a power often reverses the sign of the exponent due to the reciprocal.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Exponent Rules
Negative Exponents

Formulas

Power of a Quotient: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \)
Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)

Theorems

-

Suitable Grade Level

Grade 10