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Chapter 7: Problem 61

Simplify each of the following expressions as completely as possible. Final answers should be expressed with positive exponents only. (Assume that all variables represent positive quantities.) $$x^{-2}+y^{-1}$$

Short Answer

Expert verified
x^{-2} + y^{-1} = \frac{1}{x^2} + \frac{1}{y} \.

Step by step solution

01

Address Negative Exponents

Rewrite the terms with negative exponents as fractions. Recall that for any non-zero number a, \(a^{-n} = \frac{1}{a^n} \). Thus, \(x^{-2} = \frac{1}{x^2} \) and \(y^{-1} = \frac{1}{y} \).
02

Combine the Fractional Terms

Since there are no like terms to combine, the expression remains as a sum of fractions. Therefore, \(x^{-2} + y^{-1} = \frac{1}{x^2} + \frac{1}{y} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponents
Negative exponents can be a bit confusing at first, but they follow a simple rule. For any non-zero number, say a, raised to a negative exponent, we can write it as the reciprocal of that number raised to the positive exponent. Mathematically, this rule is expressed as: \(a^{-n} = \frac{1}{a^n}\).
Applying this rule to our exercise:
  • \(x^{-2}\) becomes \(\frac{1}{x^2}\)
  • \(y^{-1}\) becomes \(\frac{1}{y}\)
Understanding how to convert negative exponents to fractions is crucial to simplifying expressions effectively.
Fractions
Fractions are a core concept in algebra and mathematics generally. When working with algebraic expressions involving fractions, it helps to understand that a fraction represents division. For example, \(\frac{1}{x^2}\) means 1 divided by \(x^2\).
In our exercise, after converting the negative exponents, we get the fractions \(\frac{1}{x^2}\) and \(\frac{1}{y}\).
Since each term is separate and not like terms (meaning their bases are different), they stay as a sum of fractions. Therefore, the expression remains \(\frac{1}{x^2} + \frac{1}{y}\).
Positive Exponents
When simplifying algebraic expressions, it's essential to express the final answers with positive exponents. Positive exponents denote standard multiplication. For instance, \(a^2\) means \(a * a\).
In our given expression, we already converted negative exponents into positive ones by rewriting them as fractions:
  • \(x^{-2}\) became \(\frac{1}{x^2}\)
  • \(y^{-1}\) became \(\frac{1}{y}\)
By ensuring all exponents are positive, we can confirm the expression is simplified correctly. In conclusion, our given exercise simplifies to \(\frac{1}{x^2} + \frac{1}{y}\).

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