In Exercises 83–90, perform the indicated operations. Simplify the result, if possible. $$\frac{y^{-1}$ - (y + 2)^{-1}}{2}$

검증된 단계별 안내

Rewrite the expression clearly: $\frac{y^{-1} - (y+2)^{-1}}{2}$.

Express the negative exponents as fractions: $y^{-1} = \frac{1}{y}$ and $(y+2)^{-1} = \frac{1}{y+2}$, so the expression becomes $\frac{\frac{1}{y} - \frac{1}{y+2}}{2}$.

Find a common denominator for the numerator: the common denominator is $y(y+2)$, so rewrite the numerator as $\frac{(y+2) - y}{y(y+2)}$.

Simplify the numerator inside the fraction: $(y+2) - y = 2$, so the numerator becomes $\frac{2}{y(y+2)}$.

Divide the numerator by 2, which is the same as multiplying by $\frac{1}{2}$: $\frac{2}{y(y+2)} \times \frac{1}{2} = \frac{1}{y(y+2)}$. This is the simplified form.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, y^(-1) equals 1/y. Understanding this allows you to rewrite expressions with negative exponents as fractions, which is essential for simplifying the given expression.

Operations with Rational Expressions

Rational expressions are fractions involving variables. To add or subtract them, you need a common denominator. This concept is crucial for combining terms like 1/y and 1/(y+2) in the numerator before dividing by 2.

Simplifying Complex Fractions

A complex fraction has fractions in its numerator, denominator, or both. Simplifying involves rewriting the numerator and denominator as single fractions and then dividing by multiplying by the reciprocal. This process helps simplify the entire expression efficiently.

In Exercises 83–90, perform the indicated operations. Simplify the result, if possible. y−1−(y+2)−12\(\frac{y^{-1}\) - (y + 2)^{-1}}{2} 2y−1−(y+2)−1

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Negative Exponents

Operations with Rational Expressions

Simplifying Complex Fractions

In Exercises 83–90, perform the indicated operations. Simplify the result, if possible. $\(\frac{y^{-1}\) - (y + 2)^{-1}}{2}$