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Understanding Square and Cube Roots in Mathematics

Learn about square and cube roots, properties of exponents, simplifying radical expressions, complex numbers, and operations on radical expressions like addition and subtraction. Practice rationalizing denominators and using graphing calculators.

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Understanding Square and Cube Roots in Mathematics

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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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