Multiplying, Dividing, and Simplifying Radicals
Multiply square root radicals.
Simplify radicals by using the product rule.
Simplify radicals by using the quotient rule.
Simplify radicals involving variables.
Simplify other roots.
8.2
2
3
4
5
1
Product Rule for Radicals
For nonnegative real numbers a and b,
and
That is, the product of two square roots is the square root of the product, and the square root of a product is the product of the square roots.
Multiply square root radicals.
a b a b .a b a b
It is important to note that the radicands not be negative numbers in the product rule. Also, in general, .x y x y
6 11
Solution:
Find each product. Assume that 0.x
3 5
6 11
13 x
10 10
3 5
13 x
10 10
15
66
13x
100 10
EXAMPLE 1 Using the Product Rule to Multiply Radicals
Simplify radicals by using the product rule.
A square root radical is simplified when no perfect square factor remains under the radical sign.
This can be accomplished by using the product rule:
a b a b
Simplify each radical.
500
Solution:
60
17
4 15
100 5
It cannot be simplified further.
2 15
10 5
EXAMPLE 2 Using the Product Rule to Simplify Radicals
Find each product and simplify.
6 2
Solution:
10 50
6 2
10 50 500 100 5 10 5
12 2 3
EXAMPLE 3 Multiplying and Simplifying Radicals
12 5 2
12 25 2 3 5 4 10 3 4 5 10 12 50 12 25 2
60 2
The quotient rule for radicals is similar to the product rule.
Simplify radicals by using the quotient rule.
Simplify each radical.
48
3
Solution:
4
49
5
36
4
49
2
7
48
3 16 4
5
36
5
6
EXAMPLE 4 Using the Quotient Rule to Simplify Radicals
Simplify.
Solution:
8 50
4 5
8 50
4 5
502
5 2 10 2 10
EXAMPLE 5 Using the Quotient Rule to Divide Radicals
Simplify.
Solution:
3 7
8 2
3 7
8 2
21
16
21
16
21
4
EXAMPLE 6 Using Both the Product and Quotient Rules
Simplify radicals involving variables.
Radicals can also involve variables.
The square root of a squared number is always nonnegative. The absolute value is used to express this.
The product and quotient rules apply when variables appear under the radical sign, as long as the variables represent only nonnegative real numbers
2For any real number , .a a a
0, .x x x
Simplify each radical. Assume that all variables represent positive real numbers.
Solution:
6x
8100p
4
7
y
3x 23 6Since x x
8100 p 410p
4
7
y
2
7
y
EXAMPLE 7 Simplifying Radicals Involving Variables
Finding the square root of a number is the inverse of squaring a number. In a similar way, there are inverses to finding the cube of a number or to finding the fourth or greater power of a number.
The nth root of a is written
Find cube, fourth, and other roots.
.n a
In the number n is the index or order of the radical.,n a
n a
Radical sign
IndexRadicand
It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10.
Properties of Radicals
For all real number for which the indicated roots exist,
Simplify other roots.
To simplify cube roots, look for factors that are perfect cubes. A perfect cube is a number with a rational cube root.
For example, , and because 4 is a rational number, 64 is a perfect cube.
3 64 4
nand . 0n
n n n
n
a aa b ab b
bb
Simplify each radical.
Solution:
3 108
4 160
416
625
33 27 4 33 4
4 16 10 4 416 10 42 10
4
4
16
625
2
5
EXAMPLE 8 Simplifying Other Roots
Find each root.
4 81
4 81
4 81
5 243
5 243
3
3
Not a real number.
3
3
Solution:
EXAMPLE 10 Finding Other Roots
Simplify each radical.
Solution:
3 9z
3 68x
3 554t
15
3a
64
3z
22x3 63 8 x
3 3 227 2t t 3 33 227 2t t 3 23 2t t
3 15
3 64
a
5
4
a
EXAMPLE 9 Simplifying Cube Roots Involving Variables
Adding and Subtracting Radicals
Add and subtract radicals.
Simplify radical sums and differences.
Simplify more complicated radical expressions.
8.3
2
3
1
Add and subtract radicals.
We add or subtract radicals by using the distributive property. For example,
8 3 368 6 3
.14 3
and 52 2 3,
32 3as well as and 2 3 .
Radicands are different
Indexes are different
Only like radicals — those which are multiples of the same root of the same number — can be combined this way. The preceding example shows like radicals. By contrast, examples of unlike radicals are
Note that cannot be simplified.35 + 5
Add or subtract, as indicated.
Solution:
8 5 2 5 3 11 12 11 7 10
8 2 5
10 5
3 12 11
9 11
It cannot be added by the distributive property.
EXAMPLE 1 Adding and Subtracting Like Radicals
Simplify radical sums and differences.
Sometimes, one or more radical expressions in a sum or difference must be simplified. Then, any like radicals that result can be added or subtracted.
Add or subtract, as indicated.
Solution:
27 12 5 200 6 18332 54 4 2
3 3 2 3
5 3
5 100 2 6 9 2
5 100 2 6 9 2
50 2 18 2
32 2
3 332 27 2 4 2
3 32 3 2 4 2
3 36 2 4 2
310 2
EXAMPLE 2 Simplifying Radicals to Add or Subtract
Simplify more complicated radical expressions.
When simplifying more complicated radical expressions, recall the rules for order of operations.
A sum or difference of radicals can be simplified only if the radicals are like radicals. Thus, cannot be simplified further.
5 3 5 4 5, but 5 5 3
Simplify each radical expression. Assume that all variables represent nonnegative real numbers.
7 21 2 27
7 21 2 27
147 2 27
49 3 2 27
49 3 2 27
7 3 2 27
7 3 2 3 3
7 3 6 3
13 3
6 3 8r r
6 2 2r r
6 3 2 2r r
18 2 2r r
9 2 2 2r r
3 2 2 2r r
5 2r
Solution:
EXAMPLE 3 Simplifying Radical Expressions
Simplify each radical expression. Assume that all variables represent nonnegative real numbers.
2y 72 18y
29 8 9 2y y
23 8 3 2y y
23 2 2 3 2y y
26 2 3 2y y
6 2 3 2y y
3 2y
3 2y
3 33 3 5 2 3x x x x
3 34 481 5 24x x
3 33 33 327 3 5 8 3x x x x
3 33 3 10 3x x x x
313 3x x
Solution:
EXAMPLE 3 Simplifying Radical Expressions (cont’d)
Rationalizing the Denominator
Rationalize denominators with square roots.
Write radicals in simplified form.
Rationalize denominators with cube roots.
8.4
2
3
1
Rationalize denominators with square roots.
It is easier to work with a radical expression if the denominators do not contain any radicals.
1 1 2
22 2
2
2
2.
2
This process of changing the denominator from a radical, or irrational number, to a rational number is called rationalizing the denominator.
The value of the radical expression is not changed; only the form is changed, because the expression has been multiplied by 1 in the form of
Rationalize each denominator.
Solution:
18
24618
2 6 6 18 6
2 6
18 6
12
16
8
216
2 2 2 16 2
2 2
16 2
4 4 2
3 6
2
EXAMPLE 1 Rationalizing Denominators
Write radicals in simplified form.
Conditions for Simplified Form of a Radical
1. The radicand contains no factor (except 1) that is a perfect square (in dealing with square roots), a perfect cube (in dealing with cube roots), and so on.
2. The radicand has no fractions.
3. No denominator contains a radical.
Solution:
5.
18
5
18
8
5
18
18
1 5 18
18
5 9 2
18
5 9 2
18
3 5 2
18
3 10
18
10
6
EXAMPLE 2 Simplifying a Radical
Simplify
Simplify
Solution:
1 5.
2 6
1 5
2 6
5
12
5
12
35
2 3 3
5 3
6
15
6
EXAMPLE 3 Simplifying a Product of Radicals
Simplify. Assume that p and q are positive numbers.
Solution:
5p
q
5 qp
q q
5pq
q
EXAMPLE 4 Simplifying Quotients Involving Radicals
35
7
pq
2 235
7
p q
2 25
7
7
7
p q
2 2 35
7
p q
2 25
7
p q
2 25
7
p q
Rationalize each denominator.
Solution:
35
6
3
3
2
3
3
3
3, 0
4xx
2
3
3
23
3 65
6 6
3 2
3 3
5 6
6
3 180
6
2
3
3
23
3 32
3 3
3 2
3 3
2 3
3
3 18
3
3 2 2
3 23 2
3 4
4
3
4
x
x x
3 2
3 3 3
3 16
4
x
x
23 3 2 8
4
x
x
3 23 8 6
4
x
x
3 26
2
x
x
EXAMPLE 5 Rationalizing Denominators with Cube Roots
More Simplifying and Operations with Radicals
Simplify products of radical expressions.
Use conjugates to rationalize denominators of radical expressions.
Write radical expressions with quotients in lowest terms.
8.5
2
3
1
Find each product and simplify.
Solution:
2 8 20 2 5 3 3 2 2
2 2 2 4 5
2 2 2 4 5
2 2 2 2 5
4 2 5 2
4 2 10
2 3 2 2 2 5 3 3 5 3 2 2
6 11 10 6
11 9 6
EXAMPLE 1 Multiplying Radical Expressions
Find each product and simplify.
Solution:
2 5 10 2
2 10 2 2 5 10 5 2
20 2 50 10
2 5 2 5 2 10
EXAMPLE 1 Multiplying Radical Expressions (cont’d)
Find each product. Assume that x ≥ 0.
Solution:
2
5 3 2
4 2 5 2
2 x
2
25 2 5 3 3 2
24 2 2 4 2 5 5 222 2 2 x x
5 6 5 9
14 6 5
32 40 2 25
57 40 2
4 4 x x
Remember only like radicals can be combined!
EXAMPLE 2 Using Special Products with Radicals
Using a Special Product with Radicals.
Example 3 uses the rule for the product of the sum and difference of two terms,
2 2.x y x y x y
Find each product. Assume that 0.y
Solution:
3 2 3 2 4 4y y
2 2
3 2
3 4
1
2 2
4y
16y
EXAMPLE 3 Using a Special Product with Radicals
The results in the previous example do not contain radicals. The pairs being multiplied are called conjugates of each other. Conjugates can be used to rationalize the denominators in more complicated quotients, such as
Use conjugates to rationalize denominators of radical expressions.
2.
4 3
Using Conjugates to Rationalize a Binomial Denominator
To rationalize a binomial denominator, where at least one of those terms is a square root radical, multiply numerator and denominator by the conjugate of the denominator.
Simplify by rationalizing each denominator. Assume that 0.t 3
2 55+3
2 5
2 5
2 55 2
3
2
2
3 2 5
2 5
3 2 5
4 5
3 2 5
1
3 2 5
2 5
2 5
5 3
2 5
2 2
2 5 5 6 3 5
2 5
5 5 11
4 5
5 5 11
1
5 5 11
11 5 5
Solution:
EXAMPLE 4 Using Conjugates to Rationalize Denominators
Simplify by rationalizing each denominator. Assume that 0.t
3
2 t
23
2 2
t
tt
2
2
3 2
2
t
t
3 2
4
t
t
Solution:
EXAMPLE 4 Using Conjugates to Rationalize Denominators (cont’d)
Write in lowest terms.
Solution:
5 3 15
10
5 3 3
10
3 3
2
EXAMPLE 5 Writing a Radical Quotient in Lowest Terms
Using Rational Numbers as Exponents
Define and use expressions of the form a1/n.
Define and use expressions of the form am/n.
Apply the rules for exponents using rational exponents.
Use rational exponents to simplify radicals.
8.7
2
3
4
1
Define and use expressions of the form a1/n.
Now consider how an expression such as 51/2 should be defined, so that all the rules for exponents developed earlier still apply. We define 51/2 so that
51/2 · 51/2 = 51/2 + 1/2 = 51 = 5.
This agrees with the product rule for exponents from Section 5.1. By definition,
Since both 51/2 · 51/2 and equal 5,
this would seem to suggest that 51/2 should equal
Similarly, then 51/3 should equal
5 5 5.
5 5
3 5.5.
Review the basic rules for exponents:
m n m na a a m
m nn
aa
a nm mna a
Slide 8.7-4
a1/nIf a is a nonnegative number and n is a positive integer, then
1/ .n na a
Slide 8.7-5
Define and use expressions of the form a1/n.
Notice that the denominator of the rational exponent is the index of the radical.
Simplify.
491/2
10001/3
811/4
Solution:
49 7
3 1000 10
4 81 3
Slide 8.7-6
EXAMPLE 1 Using the Definition of a1/n
Define and use expressions of the form am/n.
Now we can define a more general exponential expression, such as 163/4. By the power rule, (am)n = amn, so
333/ 4 1/ 4 3416 16 16 2 8.
However, 163/4 can also be written as
Either way, the answer is the same. Taking the root first involves smaller numbers and is often easier. This example suggests the following definition for a m/n.
1/ 4 1/ 43/ 4 3 416 16 4096 4096 8.
am/nIf a is a nonnegative number and m and n are integers with n > 0, then
/ 1/ .mmm n n na a a
Slide 8.7-8
Evaluate.
95/2
85/3
–272/3
Solution:
51/ 29 53
51/38 52
21/327 9
243
32
23
Slide 8.7-9
EXAMPLE 2 Using the Definition of am/n
Earlier, a–n was defined as
for nonzero numbers a and integers n. This same result applies to negative rational exponents.
Using the definition of am/n.
1nn
aa
a−m/nIf a is a positive number and m and n are integers, with n > 0, then
//
1.m n
m na
a
A common mistake is to write 27–4/3 as –273/4. This is incorrect. The negative exponent does not indicate a negative number. Also, the negative exponent indicates to use the reciprocal of the base, not the reciprocal of the exponent.
Slide 8.7-10
Solution:
Evaluate.
36–3/2
81–3/4
3/ 2
1
36
3/ 4
1
81
31/ 2
1
36
3
1
6
1
216
31/ 4
1
81
3
1
3
1
27
Slide 8.7-11
EXAMPLE 3 Using the Definition of a−m/n
Apply the rules for exponents using rational exponents.
All the rules for exponents given earlier still hold when the exponents are fractions.
Slide 8.7-13
Solution:
Simplify. Write each answer in exponential form with only positive exponents.
1/3 2/37 72/3
1/3
9
9
5/327
8
1/ 2 2
5/ 2
3 3
3
1/3 2 /37 7
2/3 1/39 9
5/3
5/3
27
8
51/3
51/3
27
8
5
5
3
2
1/ 2 4/ 2 5/ 23 2/ 23 3
Slide 8.7-14
EXAMPLE 4 Using the Rules for Exponents with Fractional Exponents
Simplify. Write each answer in exponential form with only positive exponents. Assume that all variables represent positive numbers.
Solution:
62/3 1/3 2a b c
2/3 1/3
1
r r
r
32/3
1/ 4
a
b
6 6 62/3 1/3 2a b c 12/3 6/3 12a b c 4 2 12a b c
2/3 1/3 3/3r 6/3r 2r
32/3
31/ 4
a
b
6/3
3/ 4
a
b
2
3/ 4
a
b
Slide 8.7-15
EXAMPLE 5 Using Fractional Exponents with Variables
Use rational exponents to simplify radicals.
Sometimes it is easier to simplify a radical by first writing it in exponential form.
Slide 8.7-17
Simplify each radical by first writing it in exponential form.
4 212
36 x
1/ 4212 1/ 212 12
1/ 63x 1/ 2x 0x x
Solution:
2 3
Slide 8.7-18
EXAMPLE 6 Simplifying Radicals by Using Rational Exponents
Solving Equations with Radicals
Solve radical equations having square root radicals.
Identify equations with no solutions.
Solve equations by squaring a binomial.
Solve radical equations having cube root radicals.
8.6
2
3
4
1
Solving Equations with Radicals.
A radical equation is an equation having a variable in the radicand, such as
1 3x or 3 8 9x x
To solve radical equations having square root radicals, we need a new property, called the squaring property of equality.
Be very careful with the squaring property: Using this property can give a new equation with more solutions than the original equation has. Because of this possibility, checking is an essential part of the process. All proposed solutions from the squared equation must be checked in the original equation.
Solve radical equations having square root radicals.
Squaring Property of Equality
If each side of a given equation is squared, then all solutions of the original equation are among the solutions of the squared equation.
Solve.
Solution:
It is important to note that even though the algebraic work may be done perfectly, the answer produced may not make the original equation true.
9 4x
229 4x
9 16x 9 169 9x
7x 7x 7
EXAMPLE 1 Using the Squaring Property of Equality
Solve.
Solution:
3 9 2x x
2 2
3 9 2x x
3 9 4x x
3 33 9 4xx x x
9x
9
EXAMPLE 2 Using the Squaring Property with a Radical on Each Side
Solution:
Solve.
4x
2 2
4x
16x
16 44 4
4x
False
Because represents the principal or nonnegative square root of x in Example 3, we might have seen immediately that there is no solution.
x
Check:
EXAMPLE 3 Using the Squaring Property When One Side Is Negative
Solving a Radical EquationStep 1 Isolate a radical. Arrange the terms so that a radical is
isolated on one side of the equation.
Solving a Radical Equation.
Step 6 Check all proposed solutions in the original equation.
Step 5 Solve the equation. Find all proposed solutions.
Step 4 Repeat Steps 1-3 if there is still a term with a radical.
Step 3 Combine like terms.
Step 2 Square both sides.
Solution:
Solve 2 4 16.x x x
22 2 4 16x x x
2 22 24 16x xx x x
44 40 16xx x 4 1
4 4
6x
4x
Since x must be a positive number the solution set is Ø.
EXAMPLE 4 Using the Squaring Property with a Quadratic Expression
Solve
Solution:
2 1 10 9.x x
222 1 10 9x x
2 10 94 4 1 10 99 10x x xx x 24 14 8 0x x
2 1 2 8 0x x
2 8 0x 2 1 0x 4x 1
2x
Since 2x-1 must be positive the solution set is {4}.
or
EXAMPLE 5 Using the Squaring Property when One Side Has Two Terms
Solve.
Solution:
25 6x x
625 66x x
2 2
25 6x x 225 12 325 256x x xx x 20 13 36x x
0 4 9x x 0 9x 0 4x
9x 4x
The solution set is {4,9}.
or
EXAMPLE 6 Rewriting an Equation before Using the Squaring Property
Solve equations by squaring a binomial.
Errors often occur when both sides of an equation are squared. For instance, when both sides of
are squared, the entire binomial 2x + 1 must be squared to get 4x2 + 4x + 1. It is incorrect to square the 2x and the 1 separately to get 4x2 + 1.
9 2 1x x
Solve.
Solution:
1 4 1x x
1 1 4x x
2 2
1 1 4x x
1 1 2 4 4x x x
224 2 4x
16 4 16x 32
4 4
4x
8x The solution set is {8}.
EXAMPLE 7 Using the Squaring Property Twice
Solve radical equations having cube root radicals.
We can extend the concept of raising both sides of an equation to a power in order to solve radical equations with cube roots.