1 2 Algebra practice part 4 E. Exponents 3 4 Positive exponents Negative exponents Examples: Examples: (convention) 3-rd power of 4, 4: base, 3: exponent In general: In general: ( x any non-zero number, ( x any number, n positive integer) n positive integer) x -1 is the inverse of x Exercises: Exercises:
5 6 Radicals Radicals Example: Example: ? 3 = 8 ? 3 = –8 • 2 3 =8: 2 is the 3-rd root (cubic root) of 8 • (–2) 3 =8: –2 is the 3-rd root of –8 • the 3-rd root of 8 is denoted by • the 3-rd root of 8 is denoted by i.e. i.e. 7 8 Radicals Radicals Example: Example: ? 4 = 16 ? 4 = –16 • 2 4 =16: 2 is a 4-th root of 16 • (–2) 4 =16: also –2 is a 4-th root of 16 • no numbers whose 4-th power equals –16 • 16 has two 4-th roots: 2 and -2 • –16 has no 4-th root • positive 4-th root of 16 is denoted by i.e. • it follows that the negative 4-th root of 16 is given by i.e.
9 10 Radicals Radicals: remarks • 16 has two 4-th roots: and • 3-rd roots are cubic roots this is a typical example of the case of an even root of a • 2-nd roots are square roots: positive number • –16 has no 4-th roots this is a typical example of the case of an even root of a • for any positive integer n: negative number • 8 has one 3-rd root: this is a typical example of the case of an odd root of a • in many cases roots have to be calculated using positive number the calculator: • –8 has one 3-rd root: ♦ this is a typical example of the case of an odd root of a ♦ … negative number 11 12 Fractional-exponent-notation for roots More general fractions as exponent Examples: Example: stands for , i.e. In general: ( x any stricly positive number, n positive integer) Exercises: In general: ( x any strictly positive number, z integer, n positive integer)
13 14 Irrational exponents Product of powers with same base Example: x 3 ⋅ x 4 can be written in a simpler form : In general (real exponents and positive bases): Exercise: 15 16 Quotient of powers with same base Power of a power Example: Example: x 5 / x 3 can be written in a simpler form : (x 3 ) 2 can be written in a simpler form : In general (real exponents and positive bases): In general (real exponents and positive bases): Exercise: Exercise:
18 Product of powers with same exponent 17 Power of a power: a special case Power of a product Example: x 3 ⋅ y 3 can be written in a different form: rational exponents for positive bases only, not valid for x= –2 (x ⋅ y) 3 can be written in a different form In general (real exponents and positive bases): Exercise: ONLY for positive x-values! 20 Sum of powers with same exponent 19 Quotient of powers with same exponent Power of a quotient Power of a sum Example: Examples: x 3 /y 3 can be written in a different form: = = = In general (real exponents and positive bases): = Exercise: In general: (x+y) r can NOT be written in a simpler form:
21 Sum of powers with same exponent 22 Rules for exponents: summary Power of a sum for all real exponents and positive bases: In general: same base: power of a power: Further examples: same exponent: applied to (square) roots: 23 24 Equations with powers: example 1 Equations with powers: example 2 Write y in terms of x if y 3 = 5 ⋅ x 2 . The volume of a cube with side x is given by V=x 3 . 1. Find the volume of a cube having side 4 cm. we have to get rid 2. What is the side of a cube having volume 729 cm 3 ? of the exponent 3 3. A first cube has side 3 cm. Find the side of a second cube, whose volume is the double of the ) 1/3 ( ( ) 1/3 y 3 = 5 ⋅ x 2 volume of the first one. Answers: y = 5 1/3 ⋅ (x 2 ) 1/3 1. 64 cm 3 2. solving x 3 =729 gives x=729 1/3 =9 (cm) Answer: y = 5 1/3 ⋅ x 2/3 3. solving x 3 =2 ⋅ 3 3 gives x=3 ⋅ 2 1/3 =3.77… ≈ 3.8 (cm)
25 E. Exponents Handbook Chapter 0: Review of Algebra 0.3 Exponents and Radicals (except: rationalizing denominators, i.e. example 3, example 6.c, problems 59-68)
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