Rationalize each denominator. a. (2 √(2))/(2 √(3)-√(8)) b. (2 √(6))/(2 √(27)-√(8)) c. (2 √(2))/(

Rationalize each denominator. a. (2 √(2))/(2 √(3)-√(8)) b....

Key Concepts

Simplification of Radical Expressions

After rationalizing the denominator, the resulting expression often requires further simplification. This involves factoring out perfect square factors from under a radical to express them in simplest form, and combining like terms where possible. Simplification ensures that the final answer is both clear and minimized.

Properties of Radicals

Understanding and applying the properties of radicals is essential for simplifying expressions involving square roots. Important properties include the product rule (?a * ?b = ?(ab)) and the ability to simplify radicals by extracting perfect squares. These properties allow one to combine and reduce radical expressions efficiently.

Rationalizing Denominators

This concept involves modifying a fraction so that no irrational numbers, such as square roots, remain in the denominator. It simplifies subsequent arithmetic operations and ensures expressions are in a standardized form. The process often involves multiplying by a strategically chosen form of 1 that removes the radical from the denominator.

Conjugate Multiplication

When faced with a binomial in the denominator that contains radicals, using the conjugate is an effective technique. The conjugate of a binomial (a + b) is (a - b), and multiplying by it transforms the denominator into a difference of squares, which simplifies to remove the radicals. This method is essential for rationalizing denominators containing two terms.

Transcript

00:01 Hello everyone, here since we want to rationalize the denominator, we need to multiply our expression above and below by the conjugate of the denominator.

00:12 So in item a, for example, we are going to multiply it by 2 square root of 3 plus square root of 8.

00:23 And the same here.

00:25 And then we find 4 square root of 6 plus 2 square root of 16 and here it will be 4 times 3 minus 8 this is equal to 4 square root of 6 plus 2 times 4 since square root of 16 is 4 and here it would be 12 minus 8 then this is a plus 4 square root of 6 and here it would be also 4 then this is equal to 2 plus square root of 6 now for we follow the same process.

01:32 We multiply it by 2 square root of 27 plus square root of 8, divided by 2 square root of 27 plus square root of 8.

01:45 Then it will be 4 times square root of 27 times 6, plus 2 times square root of 8 times 6, divided by 2 times 8, by 4 times 27 minus 8.

02:09 Then this is equals to 4 times 3 times 9 times 6.

02:19 Right, since 27 is 3 times 9.

02:25 2 times square root of 8 times 6, which is equal to 2 to the 4, time 3, right? divided by 108 minus 8.

02:43 Then this is equals to 4 times 3 3 to the square times 3 times 2 which is equal to 3 to the 4 times 2 plus 2 plus 2 divided by 100, then it will be 4 times 9 square root of 2 plus 2 times 2 which is 4 square root of 3 divided by 100 then this is equal to 9 square root of 2 plus square root of 3 divided by 25 now in idency we proceed in the same way.

03:39 We multiply it by square root of 16 plus square root of 12, divided by square root of 16 plus square root of 12.

03:50 And this is equal to 2 times square root of 2.

03:56 And here square root of 16 is 4 plus square root of 12, which is 2 times square root of 3, right? here it will be 16 minus 12 this is equals to 8 times square root of 2 plus 4 times square root of 6 divided by 4 so we have that this is equal to 2 times square root of 2 plus square root of 6 plus square root of 6 and that's it.

04:41 Here in item d we proceed again the same way.

04:45 We multiply by square root of 12 plus square root of 8.

04:50 Here the same square root of 12 plus square root of 8.

04:55 Here it will be equal to 3 times square root of 2 plus 2 square root of 3 times square root of 12, which is 2 times square root of 3 plus 2 times square root of 2 right since 8 is 2 to the cube right and here it will be 12 minus 8 now it will be equals to 6 times square root of 6 plus 6 plus 6 times square root of 6 plus 6 times times 2 plus 4 times 3 plus 4 times 2 plus 4 times square root of 6.

05:52 Then divided by 4...

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