←
Key Concepts

Key Concepts

8.1 Simplify Expressions with Roots

  • Square Root Notation
    • mm is read ‘the square root of m’
    • If n2 = m, then n=m,n=m, for n≥0.n≥0.
       The image shows the variable m inside a square root symbol. The symbol is a line that goes up along the left side and then flat above the variable. The symbol is labeled “radical sign”. The variable m is labeled “radicand”.
    • The square root of m, m,m, is a positive number whose square is m.
  • nth Root of a Number
    • If bn=a,bn=a, then b is an nth root of a.
    • The principal nth root of a is written an.an.
    • n is called the index of the radical.
  • Properties of anan
    • When n is an even number and
      • a≥0,a≥0, then anan is a real number
      • a<0,a<0, then anan is not a real number
    • When n is an odd number, anan is a real number for all values of a.
  • Simplifying Odd and Even Roots
    • For any integer n≥2,n≥2,
      • when n is odd ann=aann=a
      • when n is even ann=|a|ann=|a|
    • We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

8.2 Simplify Radical Expressions

  • Simplified Radical Expression
    • For real numbers a, m and n≥2n≥2
       anan is considered simplified if a has no factors of mnmn
  • Product Property of nth Roots
    • For any real numbers, anan and bn,bn, and for any integer n≥2n≥2
       abn=an·bnabn=an·bn and an·bn=abnan·bn=abn
  • How to simplify a radical expression using the Product Property
    1. Step 1. Find the largest factor in the radicand that is a perfect power of the index.
      Rewrite the radicand as a product of two factors, using that factor.
    2. Step 2. Use the product rule to rewrite the radical as the product of two radicals.
    3. Step 3. Simplify the root of the perfect power.
  • Quotient Property of Radical Expressions
    • If anan and bnbn are real numbers, b≠0,b≠0, and for any integer n≥2n≥2 then,
       abn=anbnabn=anbn and anbn=abnanbn=abn
  • How to simplify a radical expression using the Quotient Property.
    1. Step 1. Simplify the fraction in the radicand, if possible.
    2. Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
    3. Step 3. Simplify the radicals in the numerator and the denominator.

8.3 Simplify Rational Exponents

  • Rational Exponent a1na1n
    • If anan is a real number and n≥2,n≥2, then a1n=an.a1n=an.
  • Rational Exponent amnamn
    • For any positive integers m and n,
       amn=(an)mamn=(an)m and amn=amnamn=amn
  • Properties of Exponents
    • If a, b are real numbers and m, n are rational numbers, then
      • Product Property am·an=am+nam·an=am+n
      • Power Property (am)n=am·n(am)n=am·n
      • Product to a Power (ab)m=ambm(ab)m=ambm
      • Quotient Property aman=am−n,a≠0aman=am−n,a≠0
      • Zero Exponent Definition a0=1,a0=1, a≠0a≠0
      • Quotient to a Power Property (ab)m=ambm,b≠0(ab)m=ambm,b≠0
      • Negative Exponent Property a−n=1an,a≠0a−n=1an,a≠0

8.4 Add, Subtract, and Multiply Radical Expressions

  • Product Property of Roots
    • For any real numbers, anan and bn,bn, and for any integer n≥2n≥2
       abn=an·bnabn=an·bn and an·bn=abnan·bn=abn
  • Special Products
     Binomial SquaresProduct of Conjugates(a+b)2=a2+2ab+b2(a+b)(a−b)=a2−b2(a−b)2=a2−2ab+b2Binomial SquaresProduct of Conjugates(a+b)2=a2+2ab+b2(a+b)(a−b)=a2−b2(a−b)2=a2−2ab+b2

8.5 Divide Radical Expressions

  • Quotient Property of Radical Expressions
    • If anan and bnbn are real numbers, b≠0,b≠0, and for any integer n≥2n≥2 then,
       abn=anbnabn=anbn and anbn=abnanbn=abn
  • Simplified Radical Expressions
    • A radical expression is considered simplified if there are:
      • no factors in the radicand that have perfect powers of the index
      • no fractions in the radicand
      • no radicals in the denominator of a fraction

8.6 Solve Radical Equations

  • Binomial Squares
     (a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2
  • Solve a Radical Equation
    1. Step 1. Isolate one of the radical terms on one side of the equation.
    2. Step 2. Raise both sides of the equation to the power of the index.
    3. Step 3. Are there any more radicals?
       If yes, repeat Step 1 and Step 2 again.
       If no, solve the new equation.
    4. Step 4. Check the answer in the original equation.
  • Problem Solving Strategy for Applications with Formulas
    1. Step 1. Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for by choosing a variable to represent it.
    4. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.
  • Falling Objects
    • On Earth, if an object is dropped from a height of h feet, the time in seconds it will take to reach the ground is found by using the formula t=h4.t=h4.
  • Skid Marks and Speed of a Car
    • If the length of the skid marks is d feet, then the speed, s, of the car before the brakes were applied can be found by using the formula s=24d.s=24d.

8.7 Use Radicals in Functions

  • Properties of anan
    • When n is an even number and:
       a≥0,a≥0, then anan is a real number.
       a<0,a<0, then anan is not a real number.
    • When n is an odd number, anan is a real number for all values of a.
  • Domain of a Radical Function
    • When the index of the radical is even, the radicand must be greater than or equal to zero.
    • When the index of the radical is odd, the radicand can be any real number.

8.8 Use the Complex Number System

  • Square Root of a Negative Number
    • If b is a positive real number, then −b=bi−b=bi
      a+bia+bi
      b=0b=0 a+0·iaa+0·ia Real number
      b≠0b≠0 a+bia+bi Imaginary number
      a=0a=0 0+bibi0+bibi Pure imaginary number
      Table 8.1
    • A complex number is in standard form when written as a + bi, where a, b are real numbers.
       The diagram has a rectangle with the labels “Complex Numbers” and a plus b i. A second rectangle has the labels “Real Numbers”, a plus b i, b = 0. A third rectangle has the labels “Imaginary Numbers”, a plus b i, b not equal to 0. Arrows go from the Real Numbers rectangle and Imaginary Numbers rectangle and point toward the Complex Numbers rectangle.
  • Product of Complex Conjugates
    • If a, b are real numbers, then
       (a−bi)(a+bi)=a2+b2(a−bi)(a+bi)=a2+b2
  • How to Divide Complex Numbers
    1. Step 1. Write both the numerator and denominator in standard form.
    2. Step 2. Multiply the numerator and denominator by the complex conjugate of the denominator.
    3. Step 3. Simplify and write the result in standard form.