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Chapter 7

Chapter 7. Square Root The number b is a square root of a if b 2 = a Example 100 = 10 2 = 10 radical sign Under radical sign the expression is called radicand Expression containing a radical sign is called a radical expression .

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Chapter 7

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  1. Chapter 7 Square Root The number b is a square root of a if b2 = a Example 100 = 102 = 10 radical sign Under radical sign the expression is called radicand Expression containing a radical sign is called a radical expression. Radical expressions are 6, 5 + x + 1 , and 3x 2x - 1

  2. Cube Root The number b is a cube root of a if b3 = a Example – Find the cube root of 27 3 27 = 3 33 = 3

  3. Estimating a cellular phone transmission distance R The circular area A is covered by one transmission tower is A = R2 2 The total area covered by 10 towers are 10 R , which must equal to 50 square miles Now solve R R = 1.26, Each tower must broadcast with a minimum radius of approximately 1.26 miles

  4. Expression • For every real number If n is an integer greater than 1, then a1 n = n a Note : If a < 0 and n is an even positive integer, then a1 n is not a real number. • If m and n are positive integer with m/n in lowest terms, then a m n = n a m = ( n a ) m Note : If a < 0 and n is an even integer, then a m n is not a real number. If m and n are positive integer with m/n in lowest terms, then a - m n = 1/ a m n a = 0

  5. Properties of ExponentLet p and q be rational numbers. For all real numbers a and b for which the expressions are real numbers the following properties hold. a p . a q = a p + q Product rule a - p = 1/ a p Negative exponents a/b -p = b a p Negative exponents for quotients a p = a p-q Quotient rule for exponents a q a p q = a pq Power rule for exponents ab p= a p b p Power rule for products a p = a p Power rule for products b b p Power rule for quotients 1 2 3 4 5 6 7

  6. 7.2 Simplifying Radical Expressions Let a and b are real numbers where a and b are both defined. Product rule for radical expression (Pg – 509) = , . = Quotient rule for radical expression where b = 0 (Pg 512) =

  7. Square Root Property Let k be a nonnegative number. Then the solutions to the equation. x2 = k are x = + k. If k < 0. Then this equation has no real solutions.

  8. Using Graphing Calculator [ 5, 13, 1] by [0, 100, 10]

  9. To find cube root technologically

  10. 7.3 Operations on Radical Expressions Addition 10 + 4 = (10 + 4) = 14 Subtraction 10 - 4 = (10 - 4) = 6 Rationalize the denominator (Pg 484)

  11. Using Graphing Calculator Y1 = x2 [ -6, 6, 1] by [-4, 4, 1]

  12. Pg -522 Rationalizing Denominatorshaving square roots =

  13. 7.6 Complex NumbersPg 556 x 2 + 1 = 0 x 2 = -1 x =+ Square root property - 1 Now we define a number called the imaginary unit, denoted by i Properties of the imaginary unit i i = - 1 A complex number can be written in standard form, as a + bi, where a and b are real numbers. The real part is a and imaginary part is b

  14. Pg 513 a + ib Complex Number -3 + 2i 5 -3i -1 + 7i - 5 – 2i 4 + 6i Real part a - 3 5 -1 -5 4 Imaginary Part b 2 -3 7 -2 6

  15. Complex numbers contains the set of real numbers Complex numbers a +bi a and b real Real numbers a +bi b=0 Imaginary Numbers a +bi b =0 Rational Numbers -3, 2/3, 0 and –1/2 Irrational numbers 3 And - 11

  16. Sum or Difference of Complex Numbers Let a + bi and c + di be two complex numbers. Then Sum ( a + bi ) + (c +di) = (a + c) + (b + d)i Difference (a + bi) – (c + di) = (a - c) + (b – d)i

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