Rationalize each denominator. If possible, simplify your result. (3 √(2)-√(7))/(4 √(2)+2 √(5))

Name: Problem 69, Chapter 7: Intermediate Algebra : Concepts and Applications
Uploaded: 2020-07-23T08:59:06-08:00
Duration: 1 min 51 s
Channel: Ankit Gupta
Description: Rationalize each denominator. If possible, simplify your result. (3 √(2)-√(7))/(4 √(2)+2 √(5))

step 1 The given expression is 3 square root of 2 minus square root of 7 divided by 4 square root 2 plu

Rationalize each denominator. If possible, simplify your...

Key Concepts

Algebraic Simplification

After rationalizing the denominator, further algebraic simplification is often necessary. This may involve combining like terms or factoring, aiming to express the final result in its simplest form, which is not only neater but also easier to interpret and use in subsequent calculations.

Properties of Radicals

Understanding the properties of radicals, including how to multiply, simplify, and combine them, is essential. This knowledge allows one to simplify radical expressions accurately, ensuring that radicals are properly handled during operations like multiplication or division.

Rationalization of Denominators

Rationalization is the process of eliminating irrational numbers, particularly radicals, from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by an appropriate expression that renders the denominator rational, thereby simplifying further arithmetic or algebraic manipulation.

Conjugate Multiplication

When the denominator is a binomial containing radicals, the method of using the conjugate is particularly effective. The conjugate of a binomial expression is obtained by changing the sign between its terms, and its multiplication with the original binomial leverages the difference of squares to remove the radicals from the denominator.

Use of the Difference of Squares

The difference of squares formula states that (a+b)(a-b) = a^2 - b^2, which is fundamental in rationalizing denominators. It helps in converting the product of a binomial and its conjugate into a difference of two squares, thereby simplifying the expression by eliminating radicals.

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