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

문제 52

1장, 문제 52

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

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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.

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.

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.

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