college algebra

 Module 6:  Radical Expressions, Equations, and Functions

Root and Radicals

Square Roots: Is a radical expression that has an index of 2. Generally, the index of a square root is not written: x=x2

For any positive real number  a:

· The principal or positive square root of  a, written  a , is the positive number  b such b2 = a

Example: 36 = 6

Cubic Root: The cube root of a number  a, written a3, is the number  b such b3 = a

Example: 83= 2      since 23 = 8

Root Functions Example: 

For this function f(x) = 11x−2 find f(6)

f(6) = 11(6)−2=64=8

Rational Exponents

Exponential form of an=a1/n

· When  a is nonnegative,  n can be any index

· When  a is negative,  n must be odd

Examples:

  a)7=71/2 

  b)15xy3=15xy1/3 

Simplifying Radicals:

A  perfect power is a number or expression that can be written as an expression raised to the power that is a whole number greater than 1.

A  perfect square is a number or expression that can be written as a square of an expression. A perfect square is a perfect second power.

A  perfect cube is a number or expression that can be written as a cube of an expression. A perfect cube is a perfect third power.

Examples:

1) Perfect squares: 

1=(12),4=(22),9=(32), ……….

x2=(x)2,x4=(x2)2,x6=(x3)2……….

2) Perfect cubes:

1=(13),8=(2)3,27=(3)3, ……….

x3=(x)3,x6=(x2)3,x9=(x3)3……….

Simplify Radicals using the product rule for radicals

Product rule: for nonnegative real numbers  a and  b

abn=anbn

Example

203=13203

x73=x23x53

322=16.2=16.2=42

Simplify Radicals using the quotient rule for radicals

Quotient rule: for nonnegative real numbers  a and  b

abn=anbn,b≠0

Examples:

1) 925=925=35

2) 24x3x=24x3x3=83=2

Adding, Subtracting, and Multiplying Radicals:

Add and Subtract Radicals:

· Simplify each radical expression

· Combine like radicals (if there are any)

Examples:

1) Simplify 3+27=3+3.9=3+3.3=43

2) Simplify x2−x2y+xy=x−x2.y+xy=X−xy+xy=X

Multiply Radicals

To multiply radicals, we use the product rule, and we can often simplify the new radical.

Examples: 

1)2x(8x−50)

Begin by using the distributive property: 

2x∗8x−2x∗50=16x2−100x=4x−100x=4x−10x

2) 6x38x6

Begin by using the product rule for radicals:

6x38x6=48x9=3x16x8=3x16x8=4x43x

Dividing Radicals

Rationalize Denominators:

Multiply both the numerator and denominator of the fraction by a radical that will produce a radicand in the denominator that is a perfect power for the index.

Examples:

1. Simplify:15 

15=15∗55=525=55

2) Simplify: 16a43b3

There are no common factors in the numerator and denominator. Let’s simplify the numerator before we rationalize the denominator.

16a43b3=8a32a3b3=2a2a3b3

Since the denominator contains  b and b3 is required, we need to more factors of  b or b2. Therefore, we multiply both numerator and denominator by b23

2a2a3b3=2a2a3b3b23b23=2a2ab23b33=2a2ab23b

Solving Radical Equations

A  radical equation is an equation that contains a variable in a radicand.

Steps:

1. Rewrite the equation so that one radical containing a variable is by itself (isolated) on one side of the equation.

2. Raise each side of the equation to a power equal to the index of the radical.

3. Combine like terms.

4. If the equation still contains a term with a variable in a radicand, repeat steps 1 through 3.

5. Solve the resulting equation for the variable.

6. Check all solutions in the original equation for extraneous solutions.

Examples: 

1. x=5⟶(x)2=52

X = 25

2. x−4−6=0

x−4=6

=(x−4)2=62

X – 4 = 36

X = 40

3. For f(x) = 5x−1−3x−2, find all values of x for which f(x) = 1

1 = 5x−1−3x−2

Isolate 5x−1

Then, 5x−1=1+3x−2

=  (5x−1)2=(1+3x−2)2

5x−1=(1+3x−2)(1+3x−2)

5x−1=1+3x−2+3x−2+(3x−2)2

5x−1=1+23x−2+3x−2

2x=23x−2

x=3x−2

Square both sides: x2=(3x−2)2⟶x2=3x−2

x2−3x+2=0

(x – 2) (x – 1) = 0

x = 2  or x = 1

References

Angel, A., & Rundle, D. (2018).  Intermediate algebra for college students (10th edition). Pearson.

 Module 6:  Radical Expressions, Equations, and Functions

Root and Radicals

Square Roots: Is a radical expression that has an index of 2. Generally, the index of a square root is not written: x=x2

For any positive real number  a:

· The principal or positive square root of  a, written  a , is the positive number  b such b2 = a

Example: 36 = 6

Cubic Root: The cube root of a number  a, written a3, is the number  b such b3 = a

Example: 83= 2      since 23 = 8

Root Functions Example: 

For this function f(x) = 11x−2 find f(6)

f(6) = 11(6)−2=64=8

Rational Exponents

Exponential form of an=a1/n

· When  a is nonnegative,  n can be any index

· When  a is negative,  n must be odd

Examples:

  a)7=71/2 

  b)15xy3=15xy1/3 

Simplifying Radicals:

A  perfect power is a number or expression that can be written as an expression raised to the power that is a whole number greater than 1.

A  perfect square is a number or expression that can be written as a square of an expression. A perfect square is a perfect second power.

A  perfect cube is a number or expression that can be written as a cube of an expression. A perfect cube is a perfect third power.

Examples:

1) Perfect squares: 

1=(12),4=(22),9=(32), ……….

x2=(x)2,x4=(x2)2,x6=(x3)2……….

2) Perfect cubes:

1=(13),8=(2)3,27=(3)3, ……….

x3=(x)3,x6=(x2)3,x9=(x3)3……….

Simplify Radicals using the product rule for radicals

Product rule: for nonnegative real numbers  a and  b

abn=anbn

Example

203=13203

x73=x23x53

322=16.2=16.2=42

Simplify Radicals using the quotient rule for radicals

Quotient rule: for nonnegative real numbers  a and  b

abn=anbn,b≠0

Examples:

1) 925=925=35

2) 24x3x=24x3x3=83=2

Adding, Subtracting, and Multiplying Radicals:

Add and Subtract Radicals:

· Simplify each radical expression

· Combine like radicals (if there are any)

Examples:

1) Simplify 3+27=3+3.9=3+3.3=43

2) Simplify x2−x2y+xy=x−x2.y+xy=X−xy+xy=X

Multiply Radicals

To multiply radicals, we use the product rule, and we can often simplify the new radical.

Examples: 

1)2x(8x−50)

Begin by using the distributive property: 

2x∗8x−2x∗50=16x2−100x=4x−100x=4x−10x

2) 6x38x6

Begin by using the product rule for radicals:

6x38x6=48x9=3x16x8=3x16x8=4x43x

Dividing Radicals

Rationalize Denominators:

Multiply both the numerator and denominator of the fraction by a radical that will produce a radicand in the denominator that is a perfect power for the index.

Examples:

1. Simplify:15 

15=15∗55=525=55

2) Simplify: 16a43b3

There are no common factors in the numerator and denominator. Let’s simplify the numerator before we rationalize the denominator.

16a43b3=8a32a3b3=2a2a3b3

Since the denominator contains  b and b3 is required, we need to more factors of  b or b2. Therefore, we multiply both numerator and denominator by b23

2a2a3b3=2a2a3b3b23b23=2a2ab23b33=2a2ab23b

Solving Radical Equations

A  radical equation is an equation that contains a variable in a radicand.

Steps:

1. Rewrite the equation so that one radical containing a variable is by itself (isolated) on one side of the equation.

2. Raise each side of the equation to a power equal to the index of the radical.

3. Combine like terms.

4. If the equation still contains a term with a variable in a radicand, repeat steps 1 through 3.

5. Solve the resulting equation for the variable.

6. Check all solutions in the original equation for extraneous solutions.

Examples: 

1. x=5⟶(x)2=52

X = 25

2. x−4−6=0

x−4=6

=(x−4)2=62

X – 4 = 36

X = 40

3. For f(x) = 5x−1−3x−2, find all values of x for which f(x) = 1

1 = 5x−1−3x−2

Isolate 5x−1

Then, 5x−1=1+3x−2

=  (5x−1)2=(1+3x−2)2

5x−1=(1+3x−2)(1+3x−2)

5x−1=1+3x−2+3x−2+(3x−2)2

5x−1=1+23x−2+3x−2

2x=23x−2

x=3x−2

Square both sides: x2=(3x−2)2⟶x2=3x−2

x2−3x+2=0

(x – 2) (x – 1) = 0

x = 2  or x = 1

References

Angel, A., & Rundle, D. (2018).  Intermediate algebra for college students (10th edition). Pearson.