Question

Rationalize the numerator. $$\frac{\sqrt{x+5}-\sqrt{x}}{5}$$

   Rationalize the numerator.
$$\frac{\sqrt{x+5}-\sqrt{x}}{5}$$

Precalculus

Robert Blitzer 6th Edition

Chapter 0, Problem 79

Instant Answer

Solved by Expert Erika Bustos

08/20/2021

Step 1

To do this, we multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of $\sqrt{x+5}-\sqrt{x}$ is $\sqrt{x+5}+\sqrt{x}$. So we have: $$\frac{\sqrt{x+5}-\sqrt{x}}{5} \times \frac{\sqrt{x+5}+\sqrt{x}}{\sqrt{x+5}+\sqrt{x}}$$ Show more…

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Rationalize the numerator. $$\frac{\sqrt{x+5}-\sqrt{x}}{5}$$

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Key Concepts

Rationalization

Rationalization is the process of eliminating radicals from the numerator or denominator of an expression. This technique is often used to simplify expressions and make further algebraic manipulations or limit computations easier.

Conjugate Multiplication

Multiplying an expression involving radicals by its conjugate is a common method for rationalization. The conjugate of a binomial expression involving square roots changes the sign between the two terms, and when multiplied, it helps to eliminate the square roots through the difference of squares.

Difference of Squares

The difference of squares is an algebraic identity that states a^2 - b^2 = (a - b)(a + b). When the conjugate is used to multiply a binomial involving square roots, this identity allows the square roots to be squared and thus simplifies the expression by eliminating the radical terms.

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