10.1 Radical Expressions and Graphs

Objective 1: The square root of a number N is the number which, when multiplied by itself, yields the number N.

Example: The square root of 16 is 4 because 42 is 16.

(- 4)2 = 16 so another square root of 16 is - 4.

The positive square root is the principal square root and if we were asked simply, "What is the square root of 49?", the expected answer is 7. This is usually written , where is the radical sign and 49 is called the radicand. We read this as "the square root of 49 is 7".

If we are looking for the negative square root, we write .

We can only find square roots of nonnegative numbers. This means that we cannot find because there is no number that can be squared to give – 36.

Examples:                                         A. because 122 = 144

B. because (-32)2 = 1024 (We read this as the negative square root of 1024 or the opposite of the square root of 1024.)

C. because

D. because

E. because 92 = 81

F. because (- 11)2 = 121

To find the square of an expression, multiply it by itself.

Examples: Find the square of each expression.

A. The square of 5 is 25 because 52 = 25.

B. The square of is 10 because ()2 is 10.

C. The square of is 15 because ()2 is 15.

D. The square of is 2x2 +3 because ()2 is 2x2 + 3.

Objective 2: A perfect square is number whose square root is rational (a rational number can be written as the ratio of integers.) 64 is a perfect square because .

A number whose square root is not rational is not a perfect square. 65 is not a perfect square because there is no rational number whose square is 65. We say that is irrational (an irrational number cannot be written as the ratio of integers.)

PERFECT SQUARES: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, …

** N is a positive real number that is a perfect square, then is rational.

**If N is a positive real number that is not a perfect square, then is irrational.

**If N is a negative real number, is not real.

**CAUTION** because and is not real.

Examples: A. is irrational because 27 is not a perfect square.

B. is rational because 36 is a perfect square.

C. is not real because -27 is not positive.

Objective 3: The cube root of a number, N, is the number which, when cubed, yields N.

The cube root of 8 is 2, written because 23 = 8.

In the expression , 3 is called the index or order. (In square roots the index is two and when a radical is written without an index, the 2 is understood.)

The nth root of a, written , is the number which, when multiplied by itself n times, yields a.

PERFECT CUBES: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …

PERFECT FOURTH POWERS: 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10 000, …

Examples: A. because 43 = 64

B. because (-3)3 = - 27

**We can take odd roots of negative numbers.

**There is only one real number cube root for each real number.

Examples: A. B.

C. is not real D.

E.

 

Objective 4: A radical expression is an algebraic expression which contains radicals, like or . To graph a radical function, make a table and plot the points from the table.

Problem: Graph

Solution: Domain: Range:

x 0 1 2 3 4 5 6 7 8

f(x) 0 1 1.4 1.7 2 2.2 2.4 2.6 2.8

 

This is a graph of the SQUARE ROOT FUNCTION. This is one of several basic functions that you will see repeatedly in other math courses so you should remember what it looks like.

Problem: Graph

Solution: Domain: Range:

x -8 -6 -4 -1 0 1 4 6 8 10

f(x) -2 -1.8 -1.6 -1 0 1 1.6 1.8 2 2.2

Problem: Graph

Solution: Domain: Range:

x 3 5 7 9 11

f(x) 0 1.4 2 2.4 2.8

Problem: Graph:

Solution: Domain: Range:

 

 

Problem: Graph on the same set of axes.

Solution: Which is which??? And why do they look similar??

 

Objective 5: The square root function is what is called a piecewise defined function and its formal definition involves absolute value

   For any real number a, .

Examples:

 

We often encounter roots other than square roots, and we can generalize the definition of these nth roots.

In general, if n is even and if n is odd.
  • Examples:
  • because
  • because (X8)3 = X24
  • because (y3)6 = y18
  • ASSIGNMENT: PAGES 637 – 9 problems 7 – 33, 39 – 49, 63 – 87, 91,
  • 95 – 107 (0dds) and 51 – 56 (all)
  •  

    10.2 Rational Exponents

    • It is sometimes useful to use exponential form so it is necessary to be able to change between rational and exponential form.

    Objective 1: if is real.

    Examples:

    not real

    Objective 2: as long as is real and as long as a is not 0.

    Examples:

    not real

     

    Examples:

    Objective 3: Sometimes the easiest from to use is the radical form.

    Examples:

    Objective 4: The rules for exponents apply to fractional exponents as well as integer exponents.

    Examples:

    A.

    B.

    C.

    D.

    E.

    F.

    G.

    H.

    ASSIGNMENT: PAGES 646-7 problems 1 – 10 (all), 11 – 29, 33 – 43, 47 – 91 (odd)

     

    10.3 Simplifying Rational Expressions

    Objective 1: When radicals have the same index, the radicands may be multiplied together. The

    index does not change. as long as are real.

    Examples:

    A.

    B.

    C.

    D.

    E.

    F. cannot be multiplied because the indices are not the same.

    Objective 2: When the radicals in a quotient have the same index, the radicands may be

    divided. The index does not change.

    Examples:

    Objective 3: If the following conditions are met, then a radical is in simplified form.

    1. The radicand has no factor raised to a power greater than or equal

    to the index.

    2. The radicand has no fractions.

    3. No denominator contains a radical.

    4. Exponents in the radicand and the index of the radical have no

    common factor.

    Examples:

    To simplify , look for the largest factor of a that is a perfect square.

    To simplify , look for the largest factor of a that is a perfect cube.

    To simplify , look for the largest factor of a that is a perfect fourth power.

    Examples:

    Examples:

    Objective 4: We often encounter radicals whose indices are not the same. We will

    write these in exponent form and simplify them using the rules of exponents.

    Examples:

    Objective 5: The Pythagorean formula states that in a right triangle, the sum of the squares

    of the legs is equal to the square of the hypotenuse. (The hypotenuse is the longest of

    the three sides and is opposite the right angle.)

                                                c (hypotenuse)

                        a

    b

     c2 = a2 + b2

    If we know the lengths of any two sides of a right triangle, we can use the

    Pythagorean Theorem to find the third side.

    Problem: In the triangle drawn above, let a = 8 and let b = 14. Find c.

    Solution: c2 = a2 + b2 c2 = 82 + 142 = 64 + 196 = 260

    c =

    Problem: In the triangle drawn above, let a = 4 and let c = 6. Find b.

    Solution: c2 = a2 + b2

    62 = 42 + b2

    36 = 16 + b2

    20 = b2

    b =

    Objective 6: Given any two points in the Cartesian plane , it is

    possible to find the distance between the points using the distance formula.

    Problem: Find the distance between the points (2, -1) and (5, 3).

    Solution: Let = (2, -1) and let = (5, 3)

    *** It does not matter which point is chosen as point 1 and which is chosen as point 2 – but it is important to be consistent when "plugging-in" to the formula.

    Problem: Find the distance between the points (-3, 2) and (0, -4).

    Solution: Let = (-3,2) and let = (0, -4)

    It is customary to leave the answer in radical form unless instructed to do otherwise.

     

    ASSIGNMENT: PAGES 656 – 660 problems 7 – 19, 23 – 57, 61 – 103, 109 – 113, 117 – 129, odd

     

    10.4 Adding and Subtracting Radical Expressions

    Objective1: Radical expressions are algebraic expressions which contain radicals.

    Objective 2: To simplify radical expressions that involve addition and subtraction of

    radicals, we treat the radicals as the bases of like terms. To be like terms, the

    radicals must have the same index and also the same radicand. The coefficients

    of like radicals may then be combined using the rules for signed numbers.

    Examples: A.

    B.

         

    C.

        

    D.

    E.

        

    F.

       

    G.

        

    H.

       

    *****Sums and differences cannot be taken from under radicals.

    *****If the index is not two, it must be written in its proper place.

    ASSIGNMENT: PAGES 664 – 5 problems 5 – 53 (odd)

     

    10.5 Multiplying and Dividing Radical Expressions

    Objective 1: Radical expressions may be multiplied in a manner similar to polynomials

    by using the distributive property and the FOIL method.

    Examples: A.

    B.

    C.

    D.

    E.

    This is a special product and could have been multiplied as the first term

    squared minus the second term squared.

    F.

    This is a special product and could have been multiplied as the first term

    squared minus twice the product of the terms plus the second term squared.

    G.

    Objective 2: For a radical to be in simplified form, it must not have a radical in the denominator.

    The process of removing the radical from the denominator is called rationalizing the

    denominator.

    If the denominator contains a single square root, multiply the numerator and denominator by

    the appropriate factor so that the denominator becomes the square root of a perfect square.

    Simplify the resulting fraction.

    Examples: A.

    B.

    C.

    D.

    E.

    F.

    G.

    If the denominator contains a cube root, multiply by the appropriate factor to have the cube root of a perfect cube and then simplify.

    H.

    If the denominator contains a fourth root, multiply by the appropriate factor to have the fourth root of a perfect fourth power and then simplify. Follow the same process for higher order roots.

    I.

    J.

    Objective 3: If the denominator contains a binomial, then the radical is eliminated by multiplying the numerator and denominator by the conjugate of the denominator. The product of the conjugates is a special product (the difference of two squares) and will yield an integer.

    The conjugate of is and the product of the conjugates is 25 – 2 = 23

    The conjugate of is and the product of the conjugates is 2 – 100 = -98

    (first term squared minus second term squared)

    Examples: Rationalize the denominator.

    A.

    B.

    Objective 4: Radical quotients always need to be in lowest terms.

    Examples: A.

    B.

    We can NEVER cancel terms, we must always factor and then cancel!

    Whatever is done to one part of a fraction must be done to the other part as well.

    In some applications, it is necessary to rationalize the numerator of a fraction. Do this by multiplying both the numerator and the denominator of the fraction by the conjugate of the numerator.

    Example: Rationalize the numerator of the fraction.

     

    ASSIGNMENT PAGES 672 – 675 problems 7 – 81, 85 – 99. 103 – 113, 125, 127 (odds)

    PAGE 676 SUMMARY EXERCISES problems 1 - 25

     

    10.6 Solving Equations with Radicals

    A radical equation is an equation that contains a radical. To eliminate the radical, it must first be isolated and then both sides of the equation must be raised to the power that will eliminate the radical.

    Power Rule for Solving an Equation with Radicals: If both sides of an equation are raised to the same power, all solutions of the original equation are also solutions of the new equation.

    All solutions must be checked in the original equation. Any solutions that do not check are called extraneous solutions. When listing the solutions of a radical equation, the extraneous solutions should be excluded.

    Problem: Solve the equation

    Solution: Square both sides

    5x = 15

    x = 3

    Check:

    4 = 4 the answer checks

    Problem: Solve the equation

    Solution: Isolate the radical

    Square both sides

    5x = 1

    x = 1/5

    Check:

    4 = 0 false so this solution is extraneous.

    The original equation has no solutions.

    Problem: Solve the equation

    Solution:

    x2 + 5x = 0

    x(x + 5) = 0

    x = 0 or x = -5 (extraneous)

    The only solution is x = 0.

    Problem: Solve

    Solution:

    p = 0 or p = -2 (extraneous)

    The only solution is p = 0.

    If the equation contains more than one radical, isolate and eliminate them one at a time.

    Problem: Solve

    Solution:

    (x – 3)(x + 1) = 0

    x = 3 (extraneous) or x = -1

    The solution to the original equation is x = -1.

    If the radical has an index greater than 2, raise both sides to whatever power is necessary to eliminate the radical.

    Problem: Solve

    Solution: Cube both sides

    2x + 7 = 3x – 2

    9 = x (This answer checks.)

    The same process is used to solve formulas that contain radicals.

    Problem: Solve for C:

    Solution:

    A similar process is used to solve equation involving rational exponents.

    Problem: Solve

    Solution:

    4w2 + 12w + 8 = 0

    w2 + 3w + 2 = 0

    (w + 2)(w + 1) = 0

    w = -2 or w = -1 Both answers check.

     

    ASSIGNMENT PAGES 682 – 3 problems 7 – 57 (odd) 63, 65, 67, 69

     

    10.7 Complex Numbers

    Earlier, we agreed that the square roots of negative numbers were not real because no number can be squared to yield a negative number.

    In engineering and mathematics applications, it is often necessary to find square roots of negative numbers and the imaginary operator, i allows us to do that.

    Objective 1: Using the imaginary operator, i, we can find square roots of negative numbers.

    Examples: A.

    B.

    C.

    The imaginary operator must be removed before any multiplication or division is performed.

    Remember that i2 = -1.

    Examples: A.

    B.

    C.

    D.

    E.

    Objective 2: The set of complex numbers is the set which includes the real numbers and the imaginary numbers. A complex number can be written in standard form a + bi, where a is the real part and b is the real coefficient of the imaginary part.

    To combine complex numbers using addition and subtraction, simply collect like terms.

    Examples: A. ( -1 – 8i) + (9 – 3i) = (-1 + 9) + (-8 –3)i = 8 – 11i (complex)

    B. (-3 + 2i) + (1 – 3i) + (-7 – 5i) = -9 – 6i

    C. (-1 + 2i) – (4 + 2i) = (-1 - 4) + (2 – 2)i = -5 + 0i = -5 (real)

    D. (-10 + 6i) – (-10 + 10i) = 0 – 4i = -4i (imaginary)

    ***The sum or difference of complex numbers may be real or it may be imaginary

    or it may be complex.

    Objective 4: Complex numbers can be multiplied using the FOIL method.

    Examples: A. 6i(4 + 3i) = 24i + 18i2 = 24i – 18 = -18 + 24i

    B. (6 – 4i)(2 + 4i) = 12 + 24i – 8i – 16i2 = 12 + 16i + 16 = 28 + 16i

    C. (3 + 2i)(3 + 4i) = 9 + 12i + 6i + 8i2 = 9 + 18i – 8 = 1 + 18i

    Objective 5: Since , we must eliminate any i terms when they appear in the denominator of a fraction. This is accomplished by multiplying the numerator and denominator by the complex conjugate.

    Examples: A.

    B.

    ***The product of complex conjugates is the real part squared PLUS the real coefficient of the imaginary part squared.

    Objective 6: Any power of I greater than 1 must be simplified.

    I1 = 1 i2 = -1 i3 = i (i2) = -i i4 = i2(i2) = (-1)(-1) = 1

    I5 = 1 i6 = -1 i3 = -i i8 = 1

    … i36 = 1 i224 = 1

    i raised to any power that is a multiple of four is 1.

    ASSIGNMENT PAGE 690 problems 7 – 23, 27 – 37, 41 – 55, 59 – 67, 73 – 81, 95, 97 (odds)