Simplifying Radicals

2 Key excerpts on "Simplifying Radicals"

• 

  eBook - ePub

  HP Prime Guide Algebra Fundamentals

  HP Prime Revealed and Extended

  • Larry S Schroeder(Author)
  • 2017(Publication Date)
  • Larry Schroeder

    (Publisher)

  Simplifying Radical Expressions can be used for other simplification as well. For example, one use is to determine equivalent radical expressions when operations of multiplication and division do not have the same index.

  What we do is follow the rule. First converting the radicals to exponents; adding the exponents for multiplication, subtracting the exponents for division; and then converting the exponent result back to a radical. See the following illustration.

  TI-Nspire CAS Simplifying Radical Expressions – Different Indices

  Begin by selecting the CAS key on the HP Prime. If the CAS view of the screenshot has computations, clear the history first. To clear the history, press the Clear key.

  Key in as shown. As has been case, rational exponent form is used in both results. For the second part of the illustration it was necessary to change the input to rational exponent subtraction as well. The manual solutions has part (a) and part (b) final rational exponent as the product of the exponents showing in the calculator screen.

  Rationalizing Denominators

  The final step in simplify radicals is to have to no fractions in the denominator of the radical. This translates into have no radicals in the denominator of the quotient of two roots. Each of these situations may allow us to simplify some radicals, but to completely accomplish this final step; it is often necessary to rationalize the denominator.

  With the Properties of Radicals, the root of a quotient can be expressed as the quotient of two roots. We can simplify the root of a quotient if the denominator is a perfect nth powers by expressing as the quotient of two roots and simplifying the denominator root and if possible the numerator root.

  The reverse is also true; the quotient of two roots can be expressed as the root of a quotient. If simplification is possible, express the quotient of two roots as the root of a quotient and simplify.

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  eBook - ePub

  Elementary Algebra

  • Toby Wagner(Author)
  • 2021(Publication Date)
  • Chemeketa Press

    (Publisher)

  Now that you’re familiar with the product and quotient properties of square roots, it’s time to explore irrational square roots further. At the beginning of this section, we reviewed the concept of perfect squares. Now we’ll give a more restrictive definition for perfect squares.

  Perfect squares are real numbers that are the squares of rational numbers. The numbers 25 and are examples of perfect squares because 25 = 52 and because 5 and are both rational numbers. The number 2 is not a perfect square because 2 and V2 is not a rational number. We saw other irrational square roots earlier in Examples 2 through 4. Any square root whose radicand is not a perfect square is an irrational number.

  When simplifying irrational square roots with a calculator, the closest we can come to the value of the square root is a decimal approximation, as in Example 3. However, it’s often better to give an exact value. When a square root is irrational, we can’t write an exact value without a radical sign, but in many cases, we can simplify the square root.

  Practice B — Answers

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  A square root is written in simplified form if the radicand does not have any perfect square factors other than 1. For example, is in simplified form because the factors of 14 are 1, 2, 7, and 14. The only factor that is a perfect square is 1. However, is not in simplified form because the factors of 8 are 1, 2, 4, and 8, and 4 is a perfect square. If an irrational square root is not in simplified form, we simplify it by rewriting the radicand as a product involving a perfect square and then applying the product rule for square roots.

  Here’s an example that illustrates this procedure.

  Example 8

  Simplify Give an exact answer.

  Solution

  Now that you’ve seen how the product property of square roots can be used to simplify irrational square roots, let’s take another look at the problem from the beginning of this section.

  Example 9

  Simplify an exact answer.

  Solution

  Sometimes a radicand may contain more than one perfect square factor.

  Example 10

  Simplify  

  Solution

  Practice C

  Now it’s your turn to simplify each square root. Give exact answers, not decimal approximations. If a square root is irrational, then the simplified expression will still contain a radical. When you are finished, check your work on the next page.

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