Chapter 6 Irrational and Complex Numbers Section 6

Chapter 6 Irrational and Complex Numbers

Section 6 -1 Roots of Real Numbers

Square Root n A square root of a number b is a solution of the equation x 2 = b. Every positive number b has two square roots, denoted √b and -√b.

Principal Square Root The positive square root of b is the principal square root n The principal square root of 25 is 5 n

Examples – Square Root n Simplify x 2 = 9 n x 2 + 4 = 0 n 5 x 2 = 15 n

Cube Root n A cube root of b is a solution of the equation x 3 = b.

Examples – Cube Root Simplify n 3√ 8 n 3√ 27 n 3√ 106 n 3√a 9 n

nth root 1. 2. 3. is the solution of xn =b If n is even, there could be two, one or no nth root If n is odd, there is exactly one nth root

Examples – nth root Simplify n 4√ 81 n 5√ 32 n 5√-32 n 6√-1 n

Radical The symbol n√b is called a radical n Each symbol has a name n n = index n √ = radical n b = radicand n

Section 6 -2 Properties of Radicals

Product and Quotient Properties of Radicals 1. n√ab = n√a · n√b 2. n√a÷b = n√a ÷ n√b

Examples Simplify n 3√ 25 · 3√ 10 n 3√(81/8) 2 n √ 2 a b 3 n √ 36 w n

Rationalizing the Denominator n Create a perfect square, cube or other power in the denominator in order to simplify the answer without a radical in the denominator

Examples n Simplify √(5/3) n 4 3√c n

Theorems 1. If each radical represents a real number, then nq√b = n√(q√b). 2. If n√b represents a real number, then n√bm = (n√b)m

Examples n Give the decimal approximation to the nearest hundredth. n 4√ 100 n 3√ 1702

Section 6 -3 Sums of Radicals

Like Radicals Two radicals with the same index and same radicand n You add and subtract like radicals in the same way you combine like terms n

Examples n Simplify √ 8 + √ 98 n 3√ 81 - 3√ 24 n √ 32/3 + √ 2/3 n

Examples Simplify 5 3 2 n √ 12 x - x√ 3 x + 5 x √ 3 x n n Answer n 6 x 2√ 3 x

Section 6 -4 Binomials Containing Radicals

Multiplying Binomials You multiply binomials with radicals just like you would multiply any binomials. n Use the FOIL method to multiply binomials n

Examples Simplify n (4 + √ 7)(3 + 2√ 7) n Answer n 26 + 11√ 7 n

Conjugate Expressions of the form a√b + c√d and a√b - c√d n Conjugates can be used to rationalize denominators n

Example - Conjugate Simplify 3 + √ 5 3 - √ 5 n Answer 7 + 3√ 5 2 n

Example - Conjugate Simplify n 1 4 - √ 15 n Answer n 4 + √ 15 n

Section 6 -5 Equations Containing Radicals

Radical Equation n An equation which contains a radical with a variable in the radicand.

Solving a Radical Equation n First isolate the radical term on one side of the equation

Solving a Radical Equation - Continued If the radical term is a square root, square both sides n If the radical term is a cube root, cube both sides n

Example 1 n Solve Answer n. X = 5 n

Example 2 n Solve Answer n. X = 9 n

Example 3 n Solve Answer n X = 2/9 n

Section 6 -6 Rational and Irrational Numbers

Completeness Property of Real Numbers n Every real number has a decimal representation, and every decimal represents a real number

Remember… n A rational number is any number that can be expressed as the ratio or quotient of two integers

Decimal Representation n Every rational number can be represented by a terminating decimal or a repeating decimal

Example 1 Write each terminating decimal as a fraction in lowest terms. n 2. 571 n 0. 0036 n

Example 2 Write each repeating decimal as a fraction in lowest terms. n 0. 32727… n 1. 89189189… n

Remember… n An irrational number is a real number that is not rational

Decimal Representation Every irrational number is represented by an infinite and nonrepeating decimal n Every infinite and nonrepeating decimal represents an irrational number n

Example 3 n Classify each number as either rational or irrational √ 2 √ 4/9 2. 0303… 2. 030030003…

Section 6 -7 The Imaginary Number i

Definition i = √-1 and 2 i = -1

Definition n If r is a positive real number, then √-r = i√r

Example 1 n Simplify √-5 n √-25 n √-50 n

Combining imaginary Numbers Combine the same way you combine like terms n √-16 - √-49 n i√ 2 + 3 i√ 2 n

Multiply - Example n Simplify √-4 ▪ √-25 n i√ 2 ▪ i√ 3 n

Divide - Example Simplify n 2 3 i n 6 √-2 n

Example n Simplify √-9 x 2 + √-x 2 n √-6 y ▪ √-2 y n

Section 6 -8 The Complex Number

Complex Numbers Real numbers and imaginary numbers together form the set of complex numbers n The form a + bi, represents a complex number n

Equality of Complex Numbers a + bi = c +di if and only if a = c and b = d

Sum of Complex Numbers n (a + bi ) +(c +di ) = (a + c) + (b + d)i

Product of Complex Numbers n (a + bi )▪(c +di )= (ac – bd) + (ad + bc)i

Example 1 n Simplify (3 + 6 i) + (4 – 2 i) n (3 + 6 i) - (4 – 2 i) n

Example 2 Simplify n (3 + 4 i)(5 + 2 i) n n (3 + 4 i)2 n (3 + 4 i)(3 - 4 i)

Using Conjugates n n Simplify using conjugates 5–i 2 + 3 i

Reciprocals Find the reciprocal of 3–i n Remember… the reciprocal of x = 1/x n

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