Unit 7 Rationals and Radicals Rational Expressions Reducing/Simplification Arithmetic (multiplication and division) Radicals Simplifying Arithmetic (multiplication, division, addition, subtraction) Rationalizing denominators

Rational Expressions Definition: Fractions that contain integers in their numerator and/or denominator are called _______ numbers (this is a reminder from Unit 1) Fractions that contain ________ in their numerator and/or denominator are called rational expressions.

Rational Expressions A rational expression's denominator can never be _______. A rational expression's value is ______ when its numerator is zero, and the only way a rational expression's value can be zero is for its numerator to be ______. When a rational expression's denominator is 1, the value of the rational expression is the value of its __________. A rational expression's value is 1 when its numerator and denominator have the ________ (nonzero) value. Even if a rational expression's numerator is zero, the first point applies: a rational expression's __________ can never be zero. A thing that is written ''0/0'' isn't a number. In particular, we aren't going to call it 1! When given as a final answer, a rational expression must be reduced to lowest terms!

Simplifying Rational Expressions Factor the numerator and denominator completely using the same factoring strategy we used in Unit 6 Factor out the GCF FIRST (section 5.5) Count the number of terms 2 Terms: (section 5.7) 3 Terms: (section 5.6) 4 Terms: (section 5.5) Cancel out factors that are common to both the numerator and denominator

Example 1 Original Rational Expression Factored the numerator and denominator Canceled out the common number factors Looking at the a’s, there are 5 in the top and 8 in the bottom --when canceled out, there will be 3 left in the bottom Looking at the b’s, there are 4 in the top and 3 in the bottom --when canceled out, there will be 1 left in the top Final answer

Example 2 Original Rational Expression Factored the numerator Factored the denominator Canceled out the common number factors Looking at the m’s, there are 4 in the top and 3 in the bottom—when canceled out, there will be 1 left in the bottom. Final Answer

Example 3 nor does it equal

Example 4 Original Rational Expression Factored the numerator using the techniques from Sect 5.6 Factored the denominator Canceled out the common factors (2x-1) Final Answer

Multiplying Rational Expressions Our Plan of Attack Factor all the numerators and denominators Cancel out factors common to the numerators and denominators Multiply the numerators Multiply the denominators

Example 1 Original Multiplication Problem Final answer Factored the numerator and denominator Canceled out the common no. factors (3,5,7) Looking at the a’s, there are 11 (top) & 7 (bottom)…when canceled out, there are 4 left in top. Looking at the b’s, there are 9 (top) & 10 (bottom)…when canceled out, there is 1 left in bottom. Looking at the c’s, there are 7 (top) & 6 (bottom).. When canceled out, there is 1 left in top. Final answer

Example 2 Original Multiplication Problem Factored the trinomials using grouping (Sect. 5.6) Factored last poly as a difference of squares (5.7) Cancel out common factors Canceled out the common number factors Final Answer

Dividing Rational Expressions Our Plan of Attack: Dividing rational expressions is very much like multiplying rational expressions with one extra step KEEP – SWITCH - FLIP: Keep the first fraction, Switch to multiplication, Flip the second fraction upside down Factor all the numerators and denominators Cancel out factors common to the numerators and denominators Multiply the numerators Multiply the denominators

Example 1 Original Multiplication Problem Keep-Switch-Flip Factored the numerator and denominator Canceled out the common no. factors (3,5,7) Looking at the a’s, there are 6 (top) & 12 (bottom)…when canceled out, there are 6 left in bottom. Looking at the b’s, there are 7 (top) & 12 (bottom)…when canceled out, there is 5 left in bottom. Looking at the c’s, there are 8 (top) & 5 (bottom).. When canceled out, there is 3 left in top. Final answer

Example 2 Keep-Switch-Flip Original Multiplication Problem Keep-Switch-Flip Factored the trinomials using grouping (Sect. 5.6) Factored last poly as a difference of squares (5.7) Cancel out common factors Canceled out the common number factors Final Answer

Radical Expressions  

Radical Expressions Cube Roots

Radical Expressions  

Simplifying Radicals A radical is considered simplified when: Based on what we saw in the nth root examples, we can see one of the keys in simplifying radicals is to match the index and the radicand’s exponent. A radical is considered simplified when: Each factor in the radicand is to a power less than the index of the radical The radical contains NO fractions and NO negative numbers NO radicals appear in the denominator of a fraction

Properties of Radicals Product Rule for Radicals

Properties of Radicals Quotient Rule for Radicals

Simplifying Radicals Factor the radicand Group these factors in sets numbering the same as the index Use the Product Rule or Quotient Rule for Radicals to rewrite the expression Simplify (when the index and the radicands exponent match, the radical simplifies as an exponentless radicand)

Simplifying Radicals Example 1 Original expression Factored the radicand into perfect squares (since index = 2) Rewrote using the product rule Simplified each term where the index and radicand’s exponents matched. Combined like terms

Simplifying Radicals Original expression Factored the radicand into perfect cubes (since index =3) Rewrote using the product rule Simplified each term where the index and radicand’s exponents matched. Combined like terms

Arithmetic of Radicals To Multiply Radicals Must have same index Multiply the coefficients Multiply the radicands To Divide Radicals Divide the coefficients Divide the radicands To Add or Subtract Radicals Must have same index and same radicand Add/subtract coefficients

Example 1 Original Problem To Multiply Radicals Must have same index Indexes are equal, product rule of radicals Factored the radicand into perfect squares(since index=2) Rewrote using the product rule Simplified each term where the index and radicand’s exponents matched. Combined like terms To Multiply Radicals Must have same index Multiply the coefficients Multiply the radicands

Example 2 To Divide Radicals Must have same index Divide the coefficients Divide the radicands Original Problem Use the quotient rule to combine the two radicals Treat the radical as a grouping symbol and simplify the expression under the radical Factored the radicand into perfect cubes since index= 3 Rewrote using the product rule Simplified each term where the index and radicand’s exponents matched. Combined like terms

Example 3 To Multiply Radicals Must have the same index Multiply the coefficients Multiply the radicands Original Problem Indexes are equal, product rule of radicals Factored the radicand into perfect squares (since index=2) Rewrote using the product rule Simplified each term where the index and radicand’s exponents matched. Combined like terms

Example 4 To Add or Subtract Radicals Must have same index and same radicand Add/subtract coefficients Original Problem (it does not look like these can be combined since the radicands are not equal…but before we make a final determination, we will simplify them.) Factored the radicand into perfect squares (since index=2) Rewrote using the product rule Combined like terms

Example 5 To Add or Subtract Radicals Must have same index and same radicand Add/Subtract coefficients. Original Problem (it does not look like these can be combined since the radicands are not equal…but before we make a final determination, we will simplify them.) Factored the radicand into perfect cubess (since index=3) Rewrote using the product rule Combined like terms

Example 6 Original Problem Distribution Property to clear grouping symbols Indexes are equal, product rule of radicals Combined like terms. Combined like terms

Example 7 Original Problem Distribution Property to clear grouping symbols Indexes are equal, product rule of radicals Combined like terms. Combined like terms

Fractional Exponents  

Fractional Exponent Properties are the same as for whole number exponents. All the rules below apply.

Examples of Fractional Exponent Properties Example 1: Example 2:

Examples of Fractional Exponent Properties Example 3: Example 4:

Rationalizing Denominators Simplifying Radicals: A radical is considered simplified when: The radical contains NO fractions and NO negative numbers NO radicals appear in the denominator of a fraction The technique we use to get rid of any radicals in the denominator of a fraction is called rationalizing. To rationalizing denominators, we are going to multiply the denominator by something so that the index and the radicands exponent match meaning the radical(s) in denominator will simplify as a radicand without an exponent. Remember … if you multiply the denominator times something, you must multiply the numerator times the exact same thing!

Example 1  

Example 2  

Example 3  

Example 4  

CONJUGATES (rationalizing denominators) The previous four examples had only ONE term in the denominator. If there are two terms, there is a slightly different technique required in order to rationalize the denominators. We are going to multiply the denominator by its CONJUGATE (Remember … if you multiply the denominator times something, you must multiply the numerator times the exact same thing!)

Conjugates The conjugate of is Think: The conjugate of is Think: Notice the only difference between an expression and its conjugate is the arithmetic in the middle.

Example 5  

Example 6