Rationalize the denominator.53−1\frac{5}{\sqrt3-1}3−15

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problema 52

Capitolo 1, Problema 52

Rationalize the denominator.
$\frac{5}{\sqrt3-1}$

Guida verificata passo dopo passo

Identify the expression to rationalize: $\frac{5}{\sqrt{3} - 1}$. The goal is to eliminate the square root from the denominator.

Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{3} - 1$ is $\sqrt{3} + 1$. So multiply by $\frac{\sqrt{3} + 1}{\sqrt{3} + 1}$.

Apply the multiplication: The numerator becomes $5(\sqrt{3} + 1)$, and the denominator becomes $(\sqrt{3} - 1)(\sqrt{3} + 1)$.

Use the difference of squares formula for the denominator: $(a - b)(a + b) = a^2 - b^2$. Here, $a = \sqrt{3}$ and $b = 1$, so the denominator simplifies to $(\sqrt{3})^2 - 1^2$.

Simplify the denominator to a rational number and write the final expression as $\frac{5(\sqrt{3} + 1)}{3 - 1}$. This completes the rationalization process.

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Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals (such as square roots) from the denominator of a fraction. This is done to simplify the expression and make it easier to work with. Typically, this is achieved by multiplying the numerator and denominator by a suitable expression that removes the radical.

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Rationalizing Denominators

Conjugates of Binomials

The conjugate of a binomial expression like (√3 - 1) is (√3 + 1). Multiplying a binomial by its conjugate results in a difference of squares, which eliminates the square root terms. This property is essential for rationalizing denominators that contain binomials with radicals.

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Complex Conjugates

Difference of Squares Formula

The difference of squares formula states that (a - b)(a + b) = a² - b². This formula is used to simplify expressions when multiplying conjugates. In rationalizing denominators, it helps convert expressions like (√3 - 1)(√3 + 1) into 3 - 1 = 2, removing the radical from the denominator.

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