Section 7.1: Radicals and Rational Exponentsmeucci/Meucci_files/Math1010Chapter… · Properties of nth powers and nth roots Let a be a real number and n be an integer n 2. (1) If

Chapter 7: Radicals and Complex Numbers Lecture notes Math 1010

Section 7.1: Radicals and Rational Exponents

Definition of nth root of a numberLet a and b be real numbers and let n be an integer n ≥ 2. If a = bn, then b is an nth root of a. If n = 2, theroot is called square root. If n = 3, the root is called cube root.

Definition of principal nth root of a numberLet a be a real number that has at least one (real number) nth root. The principal nth root of a is the nth rootthat has the same sign as a and it is denoted by the radical symbol n

√a. The positive integer n is the index

of the radical, and a is the radicand.

Ex.1Examples of nth roots.

(1) 3 =√9

(2) −5 =√25

(3) 2 = 4√16

(4) 4 = 3√64

Ex.2Find each principal root.

(1)√36

(2) −√36

(3)√−4

(4) 3√8

(5) 3√−8

1


Properties of nth rootsLet a be a real number.

(1) If a is positive and n is even, then a has exactly two real nth roots, which are denoted by n√a and

− n√a.

(2) If n is odd (a is any real number), then a has one real nth root, which is denoted by n√a.

(3) If a is negative and n is even, then a has no (real) nth root.

Ex.3

(1) 81 has two real square roots:√9 = 3 and −

√9 = −3.

(2) 3√27 = 3

(3)√−25 has no real square root.

Perfect squares and perfect cubesA perfect square is an integer which is a square of an integer. A perfect cube is an integer which is a cube of aninteger.

Ex.4State whether each number is a perfect square, a perfect cube, both, or neither.

(1) 81(2) −125(3) 64(4) 32(5) 1

2


Properties of nth powers and nth rootsLet a be a real number and n be an integer n ≥ 2.

(1) If n is odd, then ( n√a)n = a.

(2) If n is even, then ( n√a)n = |a|.

Ex.5Evaluate each radical expression

(1) (√5)2

(2)√(−5)2

(3) 3√43

(4)√(−3)2

(5)√−(32)

Definition of rational exponentsLet a be a real number and let n be an integer such that n ≥ 2. If the principal nth root of a exists, then

a1n = n

√a

If m is a positive integer that has no common factor with n, then

amn = (a

1n )m = ( n

√a)m and a

mn = (am)

1n = n

√am

Rules of ExponentsLet m and n be rational numbers, and let a and b represent real numbers, variables, or algebraic expressions,a 6= 0, b 6= 0.

(1) am · an = am+n

(2) am

an = am−n

(3) (ab)m = am · bm(4) (am)n = amn

(5)(

ab

)m= am

bm

(6) a0 = 1(7) a−m = 1

am

(8)(

ab

)−m=(

ba

)m3


Ex.6Evaluate each expression.

(1) 843

(2) 25−32

(3) ( 64125 )

23

(4) −16 12

(5) (−16) 12

4


Ex.7Rewrite each expression using rational exponents.

(1) x4√x3

(2)3√x2√x3

(3) 3√

x2y

Ex.8Use the rule of exponents to simplify each expression.

(1)√

3√x

(2) (2x−1)43

3√2x−1

5


Definition of radical functionA radical function is a function that contains a radical.

Ex.9Evaluate each radical function when x = 4.

(1) f(x) = 3√x− 31

(2) g(x) =√16− 3x

Domain of a radical functionLet n be an integer, n ≥ 2.

• If n is odd, the domain of f(x) = n√x is the set of all real numbers.

• If n is even, the domain of f(x) = n√x is the set of all non-negative real numbers.

Ex.10Describe the domain of each radical function.

(1) f(x) = 3√x

(2) g(x) =√x3

6


Ex.11Find the domain of

f(x) =√2x− 1

Section 7.2: Simplifying Radical Expressions

Product and Quotient Rules for RadicalsLet u and v be real numbers, variables, or algebraic expressions. If the nth roots of u and v are real, thefollowing rules are true.

•n√uv = n

√u n√v

•n

√u

v=

n√u

n√v, v 6= 0

Ex.1Simplify each radical by removing as many factors as possible.

(1)√12

(2)√48

(3)√75

(4)√162

7


Ex.2Simplify each radical expression.

(1)√25x2

(2)√12x3

(3)√144x4

(4) 3√40

(5) 5√486x7

(6) 3√

128x3y5

(7)√

8125

(8)√56x2√8

8


Ex.3Simplify

− 3

√y5

27x3

Simplifying Radical ExpressionsA radical expression is in the simplest form if

(1) All possible nth powered factors have been removed from each radical.(2) No radical contains a fraction.(3) No denominator of a fraction contains a radical.

Ex.4Rationalize the denominator in each radical expression.

(1)√

35

(2) 43√9

(3) 83√18

9


Ex.5Rationalize the denominator in each radical expression.

(1)√

8x12y5

(2) 3

√54x6y3

5z2

Ex.6Find the length of the hypothenuse of the following right triangle

Ex.7A softball diamond has the shape of a square with 60-foot sides. The catcher is 5 feet behind home plate.How far does the catcher have to throw to reach second base?

10


Section 7.3: Adding and Subtracting Radical Expressions

Like RadicalsTwo or more radical expressions are like radicals if they have the same index and the same radicand.

Ex.1Simplify each radical expression by combining like radicals

(1)√7 + 5

√7− 2

√7

(2) 3 3√x+ 2 3

√x+√x− 8

√x

(3)√45x+ 3

√20x

(4) 5 3√x− x

√4x

(5) 3√6x4 + 3

√48x− 3

√162x4

11


Ex.2Simplify

√7− 5√

7

Section 7.4: Multiplying and Dividing Radical Expressions

Ex.1Find each product and simplify

(1)√6 ·√3

(2) 3 3√5 · 3√16

(3)√3(2 +

√5)

(4)√2(4−

√8)

(5)√6(√12−

√3)

12


Ex.2Find the product and simplify

(1) (2√7− 4)(

√7 + 1)

(2) (3−√x)(1 +

√x)

Ex.3Find each conjugate of the expression and multiply the expression by its conjugate

(1) 2−√5

(2)√3 +√x

13


Ex.4Simplify

(1) √3

1−√5

(2)4

2−√3

(3)5√2√

7 +√2

14


Ex.5Perform each division and simplify

(1) 6÷ (√x− 2)

(2) (2−√3)÷ (

√6 +√2)

(3) 1÷ (√x−√x+ 1)

15


Section 7.5: Radical Equations and Applications

Raising each side of an equation to the nth powerLet u and v be numbers, variables, or algebraic expressions, and let n be a positive integer. If u = v, then itfollows that un = vn. This is called raising each side of an equation to the nth power.

Ex.1Solve √

x− 8 = 0

Ex.2Solve √

3x+ 6 = 0

16


Ex.3Solve

3√2x+ 1− 2 = 3

Ex.4Solve √

5x+ 3 =√x+ 11

17


Ex.5Solve

4√3x+ 4

√2x− 5 = 0

Ex.6Solve √

x+ 2 = x

18


Ex.7Solve √

3x+ 1 = 2−√3x

Section 7.6: Complex Numbers

The square root of a negative numberLet c be a positive real number. Then the square root of −c is given by

√−c =

√c(−1) =

√c√−1 = (

√c)i

Ex.1Write each number in i-form.

(1)√−36

(2)√− 16

25

(3)√−54

(4)√−48√−3

19


Ex.2Perform each operation.

(1)√−9 +

√−49

(2)√−32− 2

√−2

Ex.3Find each product.

(1)√−15√−15

(2)√−5(√−45−

√−4)

Definition of complex numberA number of the form a + bi, where a and b are real numbers, is called a complex number, and it is said tobe written in standard form. The real number a is called the real part and the real number b is called theimaginary part of the complex number a + bi. If b = 0, the number a + bi = a is real. If b 6= 0, the numbera+ bi is called imaginary. If a = 0, the number a+ bi is called pure imaginary number.

20


Ex.4Determine whether the complex numbers

√−9 +

√−48 and 3− 4

√3i are equal.

Ex.5Find the values of x and y that satisfy the equation

3x−√−25 = −6 + 3iy

21


Ex.6Perform each operation and write the result in standard form.

(1) (3− i) + (−2 + 4i)(2) 3i+ (5− 3i)(3) 4− (−1 + 5i) + (7 + 2i)(4) (6 + 3i) + (2−

√−8)−

√−4

Ex.7Perform each operation and write the result in standard form.

(1) (7i)(−3i)(2) (1− i)(

√−9)

(3) (2− i)(4 + 3i)(4) (3 + 2i)(3− 2i)

22


Ex.8Write each quotient of complex numbers in standard form.

(1) 2−i4i

(2) 53−2i

(3) 8−i8+i

(4) 2+3i4−2i

23

Section 7.1: Radicals and Rational Exponentsmeucci/Meucci_files/Math1010Chapter… · Properties of nth powers and nth roots Let a be a real number and n be an integer n 2. (1) If

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