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(20x^10y^2/5x^3y^7)^-2

3 min read Jun 16, 2024
(20x^10y^2/5x^3y^7)^-2

Simplifying the Expression: (20x^10y^2/5x^3y^7)^-2

This expression involves several concepts from algebra:

  • Exponents: The numbers written as superscripts (like 10, 2, 3, and 7) indicate how many times a base is multiplied by itself.
  • Fractions: The expression is written as a fraction.
  • Negative Exponents: A negative exponent indicates the reciprocal of the base raised to the positive version of the exponent.

Let's break down the simplification step by step:

Step 1: Simplify inside the Parentheses

  1. Divide the coefficients: 20 divided by 5 is 4.
  2. Divide the x terms: x^10 divided by x^3 is x^(10-3) = x^7.
  3. Divide the y terms: y^2 divided by y^7 is y^(2-7) = y^-5.

This simplifies the expression inside the parentheses to 4x^7y^-5.

Step 2: Apply the Negative Exponent

  1. Reciprocate the entire expression: This means we flip the fraction.
  2. Change the sign of the exponent: The exponent -2 becomes 2.

Now the expression becomes: (1 / 4x^7y^-5)^2.

Step 3: Simplify the Expression Further

  1. Distribute the exponent: The exponent 2 applies to both the numerator and the denominator of the fraction.
  2. Simplify the terms: Remember that (x^m)^n = x^(m*n).

This results in (1^2) / (4^2 * x^(72) * y^(-52)).

Step 4: Final Simplification

  1. Calculate the powers: 1^2 = 1, 4^2 = 16, x^(72) = x^14, and y^(-52) = y^-10.
  2. Rewrite the negative exponent: y^-10 = 1/y^10.

This gives us the final simplified expression: 1 / (16x^14y^10).

Therefore, (20x^10y^2/5x^3y^7)^-2 simplifies to 1/(16x^14y^10).

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