Advanced Algebra Notes
Chapter 7 – Powers, Roots, and Radicals
Section 7.1 – nth Roots and Rational Exponents
Objective:
 Evaluate nth roots of real numbers using both radical notation and rational
exponent notations
Vocabulary:
Index –
Index
𝑛
√𝑎
radicand
Review Roots
√12
= √2 ∙ 2 ∙ 3
3
3
√8 = √ 2 ∙ 2 ∙ 2
4
√16
=
=
√−12 = √− |12
3
=
3
√− 8 = √−| 8
4
= 𝑖√12 =
3
= √−1 ∙ 8
4
4
√−16 = √− | 16
√2 ∙ 2 ∙ 2 ∙ 2 =
=
4
= 𝑖 √16
=
Real nth Roots
If n is odd  a has one real root:
n=3
a = -125
n=3
a = 125
𝑛
√𝑎 = 𝑎
1
𝑛
3
√−125 = −5
One real root
3
√125 = +5
If n is even and a > 0  a has 2 real roots
n=2
a = 16
√16 = ±4
Two real roots
If n is even and a< 0  a has no real roots
n=2
2
a = -16
√−16 = ø
Roots are complex values
If n is even and a = 0  has 1 real root (0)
Page 1
Rational Exponents (Fractions as Exponents)
Numerator
𝐷𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
Denominator
Base
1
2
7
=
2
=
√71
√𝐵𝑎𝑠𝑒𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟
𝑥
2
3
3
= √𝑥 2
Examples:
Rewrite each of the following using rational exponent notation:
7
4
√5
√14
2
3
( √−8)
5
Evaluate Rational Exponents
 Exponent can be inside or outside the radical sign
 Use the form which is easiest to simplify
8
2
3
=
3
√8 ∙ 8 =
3
√(2 ∙ 2 ∙ 2) ∙ (2 ∙ 2 ∙ 2)
2 ∙
2
4
OR
3
( √2 ∙ 2 ∙ 2 )
22
4
2
Examples:
2
3
3
2
27
16
3
2
−
8
4
1
2
3
2
6
−
16
3
2
Page 2
Simplified Radicals
Negative Numbers
𝟑
𝟑
𝟑
√−𝟖𝟏 = √−𝟏 ∙ √𝟑 ∙ 𝟑 ∙ 𝟑 ∙ 𝟑
3
= -3 √3
Negative Exponents
√𝟐𝟓 𝒙−𝟐 𝒚−𝟑
=
=
=
𝟐𝟓
√𝒙𝟐𝒚𝟑
𝟓 ∙𝟓
√𝒙 ∙𝒙 ∙𝒚 ∙𝒚 ∙𝒚
𝟓
𝒙 𝒚 √𝒚
∙
√𝒚
√𝒚
Must rationalize the denominator
No square roots in denominator
5 √𝑦
=
5𝑥𝑦 2
Polynomials
√𝑥 2 + 6𝑥 + 9
= √(𝑥 + 3)(𝑥 + 3)
= √(𝑥 + 3)2
=x+3
Smallest Index
𝟔
√𝟐𝟕𝒙𝟑
=
𝟐∙𝟑
√𝟑 ∙ 𝟑 ∙ 𝟑 ∙ 𝒙 ∙ 𝒙 ∙ 𝒙
𝟐
𝟑
= √ √𝟑 ∙ 𝟑 ∙ 𝟑 ∙ 𝒙 ∙ 𝒙 ∙ 𝒙
= √𝟑𝒙
𝟐
Simplify – use the smallest index possible
4
√36
𝟒
√𝟗
𝟔
√𝟖
Page 3
Different Index
√𝒙 ∙
𝟑
√𝒙𝟐
=
𝒙
𝟏
𝟐
∙ 𝒙
𝟐
𝟑
= 𝒙
𝟑 𝟒
+
𝟔 𝟔
= 𝒙
𝟕
𝟔
𝟔
𝟔
= √𝒙𝟕 = 𝒙 √𝒙
Write in radical form
1
2
1
4
1
2
(5𝑎 𝑏 )
2
3
5
6
(2 𝑎 𝑏 )
Finding Real Solutions to Polynomials
 The highest degree tells the number of possible roots
 Solutions can be rational, real, or complex (imaginary)
Examples:
Solve the equation over the real numbers
2x4 = 162
2
2
x4
= 81
4
4
√𝑥 4 = √81
x
x
Divide each side by 2 (Isolate the variable)
Simplify
Find the roots of both sides
= √3 ∙ 3 ∙ 3 ∙ 3
=
3
(x – 2)3
4
OR
OR
4
√−3 ∙ −3 ∙ −3 ∙ −3
-3

2 real roots
= 10
3
3
√(𝑥 − 2)3 = √10
3
x–2
= √10
3
x
= √10 + 2
Examples:
x3 = 125
3x5 = -1
(x-2)3 = 10
(x + 4)2 = 0
x4 – 7 = 9993
Page 4
Solving Equations Over the Set of Rational, Real, and Complex Numbers
2x4 = 162
2
2
4
x
= 81
4
4
4
= √81
√𝑥
x
x
Divide each side by 2 (Isolate the variable)
Simplify
Find the roots of both sides
= √3 ∙ 3 ∙ 3 ∙ 3
=
3
OR
OR
4
x = {3, −3}
4
√−3 ∙ −3 ∙ −3 ∙ −3
-3
 Only 2 solutions
Because the exponent is 4, 4 roots are needed
SO
2x4
2
-
x4 (x2
162
2
81
=
=
0
2
Translate to standard form
Divide by 2
0
Difference of Squares
(x2 + 9) (x2 – 9) = 0
+ 9) (x + 3) (x – 3) = 0
x2 + 9 = 0
x+3=0
X2 = -9
x = -3
√𝑥 2
x
Factor
Completely
x–3=0
Set each factor = 0
Solve for x
x=3
= √−9
= ± 3i
x = {𝟑𝒊, −𝟑𝒊, 𝟑, −𝟑}
 4 Roots
± 3  Rational and Real Roots
± 3  Real Roots
± 3𝑖 𝑎𝑛𝑑 ± 3  Complex Roots
(Quadratic Formula must be used with only expressions with a degree of 2.)
Page 5
Section 7.2 – Properties of Rational Exponents
Goals:
 Use properties of rational exponents to evaluate and simplify expressions
Compute with Rational Exponents
Add/Subtract Rational Exponents
 Like combining like terms
 When a radical matches exactly, the coefficients can be added or subtracted.
Examples:
2√3 + 3√3
5√3
Both have Root 3 – combine the coefficients
4 + 3√5 + 7 + 4√5
Simplify radicals to make matching radicals if possible
4√27 + 3√3 − √48
3
3
√32𝑥 + √108𝑥
Multiply Expressions with Radicals
 Distribute or use FOIL
 Values outside the radical are multiplied and values under the radical are
multiplied
Examples:
√2 (√8 − √16) = √2 ∗ √8 − √2 ∗ √16
√16 − √2 ∗ 4
4 − 4√2
(3 + √2) (√10 + √5) = (3 ∗ √10) + (3 ∗ √5) + √2 ∗ 10 + √2 ∗ 5
3√10 + 3√5 + √2 ∗ 2 ∗ 5 + √2 ∗ 5
3√10 + 3√5 + 2√5 + 1√10
4√10 + 5√5
Page 6
Conjugates
 Have the same values but opposite operations
(√2 + 1) (√2 − 1)
(5 − √3) (5 + √3)
Divide Expressions with Radicals
 Can not have a radical in the denominator
 Multiply the numerator and denominator by the conjugate of the original
denominator
 Called rationalizing a denominator
Examples:
2+ √5
5− √2
∙
5+ √2
5+ √2
1− √2
2+ √2
Page 7
Section 7.6 Solve Radical Equations
Goals:
 Solve equations that contain radicals or rational exponents
Vocabulary
Extraneous solution –
Steps for Solving with One Radical
1. Isolate the radical on a side by itself
2. Raise each side of the equation by the appropriate power
Square root – raise to the 2nd power
Cube root – raise to the 3rd power
3. Solve the equation for x
4. Check for extraneous solutions
Examples:
√𝑥 = 5
3
√𝑥
-4=0
Check
Check
√4𝑥 − 7 + 2 = 5
Check
x – 4 = √2𝑥
Check
Page 8
Rational Exponents
 To eliminate the rational exponent – apply the exponent’s reciprocal
2𝑥
3
2
= 250
Check
Solve an Equation with 2 Radicals
 Get a radical on each side and then apply the appropriate power to both sides
√3𝑥 + 2 − 2√𝑥 = 0
Page 9
Section 7.7 Statistics and Measures of Central Tendency
Goals:
 Make a frequency chart and graph data
 Find the mean, median, mode and range of given data
 Find the standard deviation of data
 Make box-and-whisker plots
Frequency Charts
 Organizes data by making tally marks in an appropriate pre-determined span of
values
Example:
 Free-Throw Percentages
o 46, 47, 48, 50, 50, 52, 53, 55, 57, 57, 58, 60, 61, 61,
62, 63, 63, 63, 63, 63, 63, 63, 64, 64, 67, 67, 67, 69,
71, 72, 72, 72, 73, 75, 75, 75, 75, 76, 77, 78, 79, 79,
80, 81,82,82, 83, 83, 85, 89, 91, 92, 100
Make a frequency chart of the above data with the following classes:
40-49
50-59
60-69
70-79
80-89
90-100
Percentage
Tallies
40-49
50-59
60-69
70-79
80-89
90-100
Develop a bar graph to represent each class as in the above example
Page 10
Mean (average)
 sum of the numbers divided by the number of values added together
 Notation: x (read “x-bar”)
Find the mean of the free-throw percentages.
Median: (middle)
 middle number when numbers are written in order (ascending or descending)
 if there is an even number of values, the median is the mean of the 2 middle
numbers
Find the median of the free throw percentages
Mode: (most)
 number or numbers that occur most frequently in a given set of values
Find the mode of the free throw percentages
Range:
 the difference between the greatest and least values of a given set of data.
Find the range of the free throw percentages
Standard Deviation
 describes the difference (deviation) between the mean of the data and a data
value
 Notation: 𝜎 (read “sigma”)

𝜎= √
(𝑥1− 𝑥)+(𝑥2 −𝑥)+(𝑥3 −𝑥) + . . . (𝑥𝑛 −𝑥)
𝑛
Page 11
Find the standard deviation of the free-throw percentages
(46−69)2 + (47−69)2 +(48−69)2 + . . .+ (100−69)2
𝜎= √
58
Numerator ≈
𝜎 ≈ √
8660 (pre-calculated for you)
8660
57
𝜎 ≈ √152
𝜎 ≈ 12.3
When the standard deviation is greater, the data is more spread out about the mean.
Box-and-Whisker Plots
 The “box” enclose the middle half of the data set and the “whiskers” extend to the
minimum and maximum data values
The median divides the data set into two halves. The lower quartile is the median
of the lower half, and the upper quartile is the median of the upper half
Steps:
1. Order the data from least to greatest
2. Find the minimum and maximum values
3. Find the median
4. Find the lower and upper quartiles
5. Plot these five numbers on a number line
6. Draw the box, the whiskers and a line segment through the median
Outliers
 Pieces of data that are either extremely below the other pieces of data or
extremely above the other pieces of data
Page 12
Data: {10, 12, 7, 11, 20, 7, 6, 8, 9}
Find:
Mean
Median:
Mode:
Range:
Standard Deviation
Make a Frequency table: Classes 6-9, 10-12, 13-15, 16-19, 20-13
Make a Box-and-Whisker Plot
Page 13