Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0.
$\frac{a}{(\sqrt a + b) - 1}$

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Identify the expression to rationalize: $\frac{a}{\sqrt{a} + b} - 1$.

Focus on rationalizing the denominator of the fraction $\frac{a}{\sqrt{a} + b}$. To do this, multiply both the numerator and denominator by the conjugate of the denominator, which is $\sqrt{a} - b$.

Multiply numerator and denominator by $\sqrt{a} - b$: $\frac{a}{\sqrt{a} + b} \times \frac{\sqrt{a} - b}{\sqrt{a} - b} = \frac{a(\sqrt{a} - b)}{(\sqrt{a} + b)(\sqrt{a} - b)}$.

Simplify the denominator using the difference of squares formula: $(\sqrt{a} + b)(\sqrt{a} - b) = (\sqrt{a})^2 - b^2 = a - b^2$.

Rewrite the expression as $\frac{a(\sqrt{a} - b)}{a - b^2} - 1$ and then combine the terms over a common denominator if needed.

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주요 개념

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

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done by multiplying the numerator and denominator by a conjugate or an appropriate expression to create a rational denominator, simplifying the expression.

Conjugates of Binomials

The conjugate of a binomial expression like (√a + b) is (√a - b). Multiplying a binomial by its conjugate uses the difference of squares formula, which removes the square root terms in the denominator, making it rational and easier to work with.

Properties of Square Roots and Nonnegative Variables

Since variables represent nonnegative numbers, square roots like √a are defined and real. This assumption ensures that expressions involving square roots are valid and simplifies the process of rationalization without considering complex numbers.

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0.a(√a+b)−1\(\frac{a}{(\surd a+b)-1}\)