By V. A. Tkachenko

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**Extra info for A problem in the spectral theory of an ordinary differential operator in a complex domain**

**Example text**

Rules for radicals: n n n Product Rule: ͙ a и ͙ b ͙ ab n ͙a n a Quotient Rule: n ,b 0 b ͙b ᭤ Radicals usually are not left in the denominator when simplifying. Removing the radical from the denominator is called rationalizing the denominator. ᭤ If a is a real number and n is a positive integer greater than 1, then ᭤ Let ͱ n a1͞n ͙ a, where a Ն 0 when n is even. ᭤ Let a be a positive real number and let m and n be integers. Then n n am͞n ͙ am (͙ a )m. 3 Exercises ᭤Skills This set of exercises will reinforce the skills illustrated in this section.

We are often interested in finding the value of an algebraic expression for a given value of a variable. This is known as evaluating an algebraic expression and is illustrated in the next example. 12 Chapter P ■ Algebra and Geometry Review Example 1 Evaluating an Expression The number of students at an elementary school is given by 300 ϩ 20x, where x is the number of years since 2003. Evaluate this expression for x 4, and describe what the result means. ᭤Solution We substitute x 4 in the algebraic expression to obtain 300 ϩ 20x 300 ϩ 20(4) 300 ϩ 80 380.

Ab)s asbs 0 6. as asϪt, a at 0 7 Expressions Containing Rational Exponents Evaluate the following expressions without using a calculator. 1)0 ͩͪ ᭤Solution (a) Using the rules for rational exponents, (4)3͞2 (41͞2)3 23 8. (b) Using the rules for exponents, ͩͪ Ϫ2 1Ϫ2 1 32 1 Recall that 3Ϫ2 2 Ϫ2 2 2 9. 1)0 1, since any nonzero number raised to the zero power is defined to be equal to 1. 1 3 ✔ Check It Out 7: Evaluate the following expressions without using a calculator. 3 ■ Roots, Radicals, and Rational Exponents 25 Example 8 Simplifying Expressions Containing Rational Exponents Simplify and write with positive exponents.