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Understanding (x+y)^-2
The expression (x+y)^-2 represents the reciprocal of the square of the sum of x and y. Let's break this down further:
Key Concepts
- Exponent: The exponent -2 indicates that we are squaring the base (x+y) and then taking its reciprocal.
- Reciprocal: The reciprocal of a number is 1 divided by that number. So, (x+y)^-2 is equivalent to 1/(x+y)^2.
- Squaring: Squaring a term means multiplying it by itself. Therefore, (x+y)^2 = (x+y)*(x+y).
Expanding the Expression
To fully understand (x+y)^-2, it's helpful to expand the squared term:
- Expand the square: (x+y)^2 = (x+y)*(x+y) = x^2 + 2xy + y^2
- Take the reciprocal: 1/(x+y)^2 = 1/(x^2 + 2xy + y^2)
Example
Let's say x = 2 and y = 3. Then:
(x+y)^-2 = (2+3)^-2 = 5^-2 = 1/5^2 = 1/25
Applications
The expression (x+y)^-2 can appear in various contexts, including:
- Algebraic manipulations: Simplifying expressions, solving equations, and working with functions.
- Calculus: Derivatives and integrals often involve terms with negative exponents.
- Physics and engineering: Formulas and models in fields like electricity and mechanics may contain expressions with negative exponents.
Important Notes
- Restrictions: The expression (x+y)^-2 is undefined when x+y = 0, since division by zero is not allowed.
- Fractions: It is often easier to work with the expression in its fractional form: 1/(x+y)^2.
- Exponents and order of operations: Remember to follow the order of operations when evaluating expressions with exponents and other operations.
Understanding the meaning and properties of (x+y)^-2 is crucial for navigating various mathematical and scientific applications. By applying the concepts of exponents, reciprocals, and expansion, you can confidently work with this expression and its variations.