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Rationalizing Radical Expressions

Rationalizing Radical Expressions. Section 7.5. What does it mean to “rationalize the denominator?”. Recall: A rational number is a terminating or repeating decimal. We prefer to have whole numbers in the denominators. It makes them easier to combine.

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Rationalizing Radical Expressions

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  1. Rationalizing Radical Expressions Section 7.5

  2. What does it mean to “rationalize the denominator?” Recall: A rational number is a terminating or repeating decimal. We prefer to have whole numbers in the denominators. It makes them easier to combine. How can we change the denominator into a whole (rational) number?

  3. Rationalize the denominator: Relax! It is easy to change an irrational into a rational! If possible, simplify the denominator and/or numerator. Multiply the denominator by the same irrational number. Squaring a square root cancels it out. Multiply the numerator as well so the value doesn’t change.

  4. Rationalize the denominator: Relax! It is easy to change an irrational into a rational! If possible, simplify the denominator and/or numerator. Multiply the denominator by the part that is irrational. Multiply the numerator as well so the value doesn’t change.

  5. Rationalize the denominator: Relax! It is easy to change an irrational into a rational! If possible, simplify the denominator and/or numerator. When working with roots other than square roots, multiply the denominator by the remaining amount needed to cancel the denominator and make it rational. Multiply the numerator as well so the value doesn’t change. We need another

  6. Rationalize the denominator: If possible, simplify the denominator and/or numerator. When working with roots other than square roots, multiply the denominator by the remaining amount needed to cancel the denominator and make it rational. Multiply the numerator as well so the value doesn’t change. We need another

  7. What does it mean to “rationalize the denominator?” Recall: A rational number is a terminating or repeating decimal. We prefer to have whole numbers in the denominators. It makes them easier to combine. How can we change a denominator like this into a whole (rational) number?

  8. Properties and Rules for Radicals Product Rule for Radicals Quotient Rule for Radicals Like radicals Conjugates Radicals with the same radicand and index/root. We can only add/subtract like radicals. The conjugate of (a + b) is (a – b). It follows that (a + b) (a – b) = a2 –b2

  9. Rationalize the denominator: The conjugate is When working denominators of sums or differences, multiply the denominator by the conjugate to make it rational. Multiply the numerator as well so the value doesn’t change.

  10. Rationalize the denominator: The conjugate is When working denominators of sums or differences, multiply the denominator by the conjugate to make it rational. Multiply the numerator as well so the value doesn’t change.

  11. Can we also “rationalize the numerator?” Yes! The process is the same. The objective is to make the numerator rational, just as we did for the denominator of the previous examples.

  12. Solving Radical Equations Section 7.6

  13. Solving Radical Equations One radical in the problem Two radicals in the problem Only two radicals in the problem and nothing else Two radicals in the problem and other stuff not under a radical The Big Picture. There are 3 different types of problems. Opposite Operations

  14. Solving Radical Equations One radical in the problem • Get radical alone on one side of = • Apply same power as root (opposite operation) to both sides to cancel the radical • Solve for x and check solution

  15. Solving Radical Equations Two radicals in the problem Only two radicals in the problem and nothing else • Get each radical alone on one side of = • Apply same power as root (opposite operations) to both sides to cancel the radical • Solve for x and check solution

  16. Solving Radical Equations Two radicals in the problem Two radicals in the problem and other stuff not under a radical • Get one radical alone to one side of = • Get other radical and extra stuff alone on other side of = • Apply same power as root (opposite operation) to both sides to cancel one radical • Get remaining radical alone on one side of = and apply same power as root to both sides and cancel the radical • Solve for x and check solution

  17. Application: Finding the missing side of a right triangle How do find the missing side length? Hypotenuse c = a = ? 9 units b= 6 units The opposite of a square is a square root

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