review of radical expressions and equations
TRANSCRIPT
Page 1 of 22
Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize denominators. Find domain of a radical function. Evaluate expressions involving rational exponents. Solve a radical equation. Evaluate square roots or cube roots. Example: Evaluate 9
Identification/Analysis This is a square root. The number under the radical is 9. (An asterisk means multiply.) *
Solution 29 3 3= = Find a number such that the number squared is 9. This number is 3 because 93332 =∗=
Example: Evaluate 3 8
Identification/Analysis This is a cube root. The number under the radical is 8.
Solution 3 33 8 2 2= = Find a number such that the number cubed is 8. This number is 2 because 82222 . 3 =∗∗=
Page 2 of 22
Example: Evaluate 400−
Identification/Analysis This is a square root. The number under the radical is –400.
Solution 400− does not have a real number solution. Find a number such that the number squared is –400. Note that 4002020 =∗ and
4002020 =−∗− (not –400). Any real number squared is a positive number or zero.
Example: Evaluate 400−
Identification/Analysis This is a square root. The number under the radical is 400.
Solution 2400 2020
− = −= −
. Find a number such that the number squared is 400. ( ) Thus 20 20 400∗ = 220 20=
Don’t forget the negative sign in front of .
Example: Evaluate 3 27−
Identification/Analysis This is a cube root. The number under the radical is –27.
Solution ( )33 327 3
3
− = −
= −
Find a number such that the number cubed will give you –27. The number is –3 because
27)3()3()3()3( 3 −=−∗−∗−=− .
Page 3 of 22
Simplify radical expressions and Rationalize denominators
Usually to simplify means to rewrite the expression in such a way that it has as few radicals as possible, and that the expression under each radical does not contain perfect powers. Some rules for radicals are illustrated below. a. 332 = or 3)3( 2 = b. 10310*32 = c. 57565 =+
d. 357*57*5 == e. 757575 =÷=÷
Also, and 105*252 77)7( == 7)52(52 666*6 == +
Keep in mind that: 75 + does not equal 75+ , and 75 does not equal
75 .
Example: Simplify 28 .
Identification/Analysis The only thing which can be “simplified” is 28 under the square root. Can it be factored in
such a way that one of the multiples is a perfect square?
Solution
2
2
28 4*7
2 *7
2 * 7
2 7
=
=
=
=
Factor 28 so one factor is a perfect square. The radical of a product equals the product of the radicals. Simplify the perfect square.
Page 4 of 22
Example: Simplify 27x . Assume . 0≥xIdentification/Analysis The expression under the square root has a perfect square.
Solution 2 27 7
7
x x
x
=
=
The radical of a product equals the product of the radicals. Simplify the perfect square.
Example: Simplify 5k . Assume . 0≥k Identification/Analysis The radical is a square root. The expression under the square root is not a perfect square.
Solution 5 4
4
2
k k k
k k
k k
= ∗
= ∗
=
.
Factor so there are perfect squares. The radical of a product equals the product of the radicals. Simplify the perfect square.
k remains under the radical since the kexponent on is 1 which is smaller than the
index (2) of the radical Example: Simplify 87320 qk . Assume . 0 and 0 ≥≥ qk Identification/Analysis The radical is a square root. The expression under the square root is not a perfect square.
Solution 7 8 7 8
6 8
6 8
3 4
3 4
320 320
5 64
5 64
5 8
8 5
k q k q
k k q
k k q
k k q
k q k
= ∗ ∗
= ∗ ∗ ∗ ∗
= ∗ ∗ ∗ ∗
= ∗ ∗ ∗ ∗
=
Factor so there are perfect squares. The radical of a product equals the product of the radicals. Simplify the perfect squares. The radical is simplified because there are no perfect squares left under the radical and the remaining variables and numbers have exponents of one.
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Example: Simplify 3 1527k . Assume . 0≥k
Identification/Analysis This is a cube root. Each factor is a perfect cube. and 3327 = ( )3515 kk = .
Solution
( )5
3 353 3
3 1533 15
3
3
2727
k
k
kk
=
∗=
∗=
The radical of a product equals the product of the radicals. Write each factor as a perfect cube. Simplify the cube roots.
Example: Simplify 3 5464 yx . Assume . 0 and 0 ≥≥ yx
Identification/Analysis The radical is a cube root. The expression under the cube root is not a perfect cube.
Solution
3 2
3 23
3 23 333 33 3
3 233 33 3
3 53 433 54
3
3
4
4
6464
xyxy
yyxx
yyxx
yyxx
yxyx
=
∗∗∗∗=
∗∗∗∗=
∗∗∗∗=
∗∗=
The radical of a product is the product of the radicals. Factor so there are perfect cubes. Simplify the perfect cubes. Rearrange the factors. Generally the radical factor is written last The radical is simplified because there are no perfect cubes left in the radical.
Page 6 of 22
Example: Simplify tt 83 . Assume . 0≥t Identification/Analysis The expression is the product of two square roots.
Solution
62
62
24
24
8383
22
2
2
t
t
t
t
tttt
=
∗∗=
∗=
=
∗=
The product of two radicals equals the radical of their product. Multiply. The radical of the product equals the product of the radicals. Factor so there are perfect squares. Simplify the square roots and rearrange.
Example: Simplify 5
43
3+
−y
y . Assume . 3≥y
Identification/Analysis The expression is the product of two square roots.
Solution
15332
153)3(2
)5(34)3(
54
33
54
33
2
+−
=
+−
=
+∗∗−
=
+∗
−=
+∗
−
yy
yy
yy
yy
yy
The product of two radicals equals the radical of the product. Multiply. Factor the perfect square in the numerator. Distribute 3 in the denominator. Simplify the perfect square.
Page 7 of 22
Example: Simplify 3
3
6162 .
Identification/Analysis The expression is the quotient of two cube roots.
Solution
327
6162
6162
3
33
3
==
=
The quotient of two radicals equals the radical of their quotient. Divide. Simplify the perfect cube. 327 3=
Example: Simplify 5854 + .
Identification/Analysis The expression is the sum of two radicals. The radicals are like radicals. Both contain 5 .
Solution
512
5)84(5854
=
+=+
Add the like radicals.
Example: Simplify 134159133154 +++− .
Identification/Analysis The expression contains the sum of four radical terms. The like radicals can be added.
Solution
137155
13)23(15)94(
134133159154134159133154
+=
+++−=
+++−=+++−
Rearrange so like radicals are together. Combine the like radicals.
Page 8 of 22
Example: Simplify 63975 +− .
Identification/Analysis The expression contains the sum of two square roots.
Solution
722
72775
73975
7997563975
=
+−=
∗∗+−=
∗∗+−=+−
Simplify the radicals to determine if there are like radicals. Combine the like radicals.
Example: Simplify 838 −+− xx . Assume . 8≥x
Identification/Analysis The expression contains the sum of two terms containing radicals.
Solution 84838 −=−+− xxx The radicals in both terms are like radicals ( 8−x ), so the terms can be added.
Example: Simplify 1892 +++ xx . Assume 2−≥x .
Identification/Analysis The expression contains the sum of two terms.
Solution
24
232
)2(32
)2(9218922
+=
+++=
+∗++=
+++=+++
x
xx
xx
xxxx
Simplify the radicals. Combine the like radicals.
Page 9 of 22
Example: Rationalize the denominator 5
3 .
Identification/Analysis Rationalize the denominator means to find an equivalent fraction whose denominator does not
contain a radical.
Solution
553
)5(53
5553
53
2
=
=
∗∗
=
Multiply the numerator and the denominator by 5 . (Note: You can’t just square the numerator and the denominator. It will change the value of the fraction) Simplify the denominator.
Page 10 of 22
Example: Rationalize denominator 2525
+−
.
Identification/Analysis The denominator of this expression is irrational because it includes an irrational number ( 2 ).
Rationalize denominator means to find an equivalent expression, but with a rational denominator.
Solution
2 2
5 2 (5 2)(5 2)5 2 (5 2)(5 2)
5 2 5 2 ( 2)25 2
25 10 2 223
27 10 223
− − −=
+ + −
− ∗ +=
−− +
=
−=
There is a binomial in the denominator. Multiply the numerator and denominator by
( )5 2− . (The conjugate of the denominator.)
Find the product of the numerator and denominator. Combine like radicals. Note: Choosing ( )5 2− uses the formula
22( )( )a b a b a b+ − = − . In this example, 5=a and 2=b . Squaring The fraction would change the value of the fraction.
Page 11 of 22
Find the domain of a function Domain The domain is a list or set of all possible inputs that yield a real number output. There are three operations we “can’t do” with real numbers in algebra. Each of these restrict the domain.
• Can’t divide by zero. • Can’t take the square root (or any even-index radical) of a negative number. • Can’t take the logarithm of zero or a negative number.
Two common notations to write the domain are set-builder and interval notation.
1. Set-builder notation: Sets are typically written in braces { } . The notation is
{ }independent variable some property or restriction about independent variable where the vertical line is read “such that.”
Example: “All real numbers, x , less than 2.” { }2x x < Example: “All real numbers, , greater than or equal to n 4− and less than 6.” { }4 6n n− ≤ <
2. Interval notation: Parenthesis indicate the starting or ending value is not included and a square bracket indicates the starting or ending value is included. Within the parentheses or square bracket, we indicate the smallest value of x followed by a comma and then the largest value of x . The examples above are shown using interval notation.
Example: “All real numbers, x , less than 2.”
( ), 2−∞ Example: “All real numbers, , greater than or equal to n 4− and less than 6.”
[ )4,6−
Page 12 of 22
Find the domain of a radical function
Example: Find the domain of 122 −= xy .
Identification/Analysis The function contains a square root. The expression under the square root, 22 1x − , must be greater or equal to zero.
Solution 0122 ≥−x
122 ≥x
6≥x In interval notation the answer is [ )6, ∞
Isolate the term with a variable . Divide both sides by 2. (2 is positive, so don’t change the inequality sign)
Example: Find the domain of 5 3y t= − .
Identification/Analysis The function contains a square root. The expression under the square root, 5 3t− , must be greater or equal to zero.
Solution 5 3 0t− ≥
3 5t− ≥ −
53
t ≤
In interval notation the answer is 5,3
⎛ ⎤−∞⎜ ⎥⎝ ⎦.
Isolate the term with a variable. 3Divide both sides by− . (Remember when
you multiply or divide an inequality by a negative number, the inequality sign changes direction.)
Page 13 of 22
Example: Find the domain of 3 14+= xy .
Identification/Analysis The function contains cube root. The expression under the cube root can be any real number.
Solution 14x + ( )14x + can be any real number ( ),−∞ ∞
Write the expression under the radical. (The expression is called the radicand.) Since the radical is a cube root the expression can be any real number. Write the domain using interval notation.
Evaluate expressions involving rational exponents For the problems in this group, an expression containing rational exponents should be written using radical notation, and an expression containing radical notation should be written using rational exponents.
The definition of rational exponent is nn xx1
=
The definition of a negative exponent is nn
xx 1
=− .
All rules for exponents apply to rational and negative exponents. Often used rules are listed below. Assume: 0a ≠
Product Rule: m n m na a a += Power Rule: ( )nma = mna Quotient Rule: m
m nn
a aa
−=
Example: Write 37 using rational exponents.
Identification/Analysis The expression contains a cube root, which could be rewritten using rational exponents.
Solution ( )
12
12
32
3 3
3*
7 7
7
7
=
=
=
Rewrite the radical using a rational exponent. Use the power rule.
Page 14 of 22
Example: Evaluate 238 .
Identification/Analysis
The exponent is 23
, which is a rational number. Use the definition for rational exponents.
Solution 2 1
3 3
13
2
2
23
2
8 8
(8 )
( 8)(2)4
∗=
=
=
==
Use the power rule. The definition of a rational exponent is used. Simplify the cube root. Note: If you use a calculator, remember to use parentheses. Enter 8^(2/3), not 8^2/3.
Page 15 of 22
Example: Evaluate 329
−
Identification/Analysis The exponent is 3
2− , which is a negative rational number.
Solution
( )
( )
( )
32
32
12
12
3
3
3
3
1991
91
9
1
9
13127
−
∗
=
=
=
=
=
=
Use the definition of a negative exponent. Use the power rule.
Write the power 12
as square root.
Simplify the square root and raise the result to the third power. If you use a calculator, remember to use parentheses. Enter 9^(-3/2), not 9^-3/2
Page 16 of 22
Solve radical equations A radical equation is an equation containing one or more radical terms. For example, xx 2= is a radical equation. To solve means to determine all the real values which, when substituted in the equation for x , will make the statement true. All such real values should be included in the answer. Note that 020 ∗= is true. Hence, is a solution of the equation 0x = xx 2= . It turns
out that 41
=x is also a solution of xx 2= 1 1verify that 24 4
⎛ ⎞= ∗⎜ ⎟⎟ . This second solution is often ignored. ⎜
⎝ ⎠To find all the solutions, follow the steps for solving radical equations given below. For equations containing one radical, the steps are:
1. Isolate the radical. 2. Square both sides of the equation 3. Solve the resulting equation which no longer contains radicals. This equation is often linear, quadratic, or rational.
x4. Check the answers. (It is possible that some values may be in the solution set of the resulting equation, but will not make the original equation true.)
For equations with two radicals:
1. Isolate one of the radicals. 2. Square both sides of the equation. 3. Combine like terms. 4. Now the equation either has no radicals, or just one radical term.
a. If there are no radicals follow steps 3 and 4 under equations containing one radical. b. If there is one radical follow steps 1 through 4 under equations containing one radical.
Page 17 of 22
Example: Solve the equation 62 =+x
Identification/Analysis This is an equation (contains an equal sign). Use the steps for equations containing one radical.
Solution 2 2
2 6
( 2) 62 36
2 2 36 234
x
xx
xx
+ =
+ =+ =
+ − = −=
The radical is isolated. Square both sides. Solve the resulting linear equation.
Check 34 2 6
36 66 6
+ =
==
Substitute 34x = into the original equation and simplify the results. The statement is true so the solution is
34x = . Example: Solve the equation 62 −=+x
Identification/Analysis This is an equation (contains an equal sign). Use the steps for equations containing one radical.
Solution
( ) ( )2 2
2 6
2 6
2 3634
x
x
xx
+ = −
+ = −
+ ==
The radical is isolated. Square both sides. Solve the resulting linear equation.
Check 34 2 6
36 66 6
+ = −
= −= −
Substitute 34x = into the original equation and simplify the results. The statement is false, so 34x = is not a solution. Note: The square root of any real number can’t be negative. Hence, 62 −=+x can’t be true.
Page 18 of 22
Example: Solve the equation 2 1 4 2 1 40x x+ − = . −
Identification/Analysis This is an equation (contains an equal sign). It contains two radical terms.
Solution
( )22
2 1 4 2 1 40
5 2 1 40
2 1 8
2 1 8
2 1 6465 32.52
x x
x
x
x
x
x
− + − =
− =
− =
− =
− =
= =
Add like radicals. Follow the steps for equations containing one radical. Divide both sides by 5. The radical is isolated. Square both sides. Solve the resulting linear equation.
Check ( ) ( )
( )
65 652 22 1 4 2 1 40
65 1 4 65 1 40
64 4 64 408 4 8 40
8 32 4040 40
− + − =
− + − =
+ =
=
+ ==
Substitute 652x = into the original equation
and simplify the results. The statement is true so 65
2x = is a solution to the equation.
+
Page 19 of 22
Example: Solve the equation 02 =− xx
Identification/Analysis This is an equation (contains an equal sign). It contains one radical term.
Solution
( ) ( )2 2
2
2
2 0
2
2
44 0
(1 4 ) 0
x x
x x
x x
x xx x
x x
− =
=
=
=
− =− =
0=x or 0)41( =− x 0x = or 4 1x− = −
0x = or 14
x =
Isolate the radical. Square both sides; remember to square each factor. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible answers.
Check
Check 0x = : ( )0 2 0 00 0
− =
=
Check 14
x = : 1 12 04 4
⎛ ⎞− =⎜ ⎟⎝ ⎠
1 2 044
1 1 02 2
0 0
− =
− =
=
Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Both answers check.
Page 20 of 22
Example: Solve the equation 33 −=+ xx
Identification/Analysis This is an equation (contains an equal sign). It contains one radical term.
Solution 33 −=+ xx
22)3(3 −=+ xx
963 2 +−=+ xxx 0672 =+− xx( )( )1 6 0x x− − =
1x 0− = or 6 0x − = 1x = or 6x =
The radical is isolated. Square both sides. Note: . 222 3)3( −≠− xx This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible solutions.
Check Check 1x = : 1 3 1 3+ = −
4 22 2= −= −
Check 6x = : 6 3 6 3+ = −
9 33 3==
Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Notice 1x = does not yield a true statement, while 6x = yields a true statement. The solution is 6x = .
Page 21 of 22
Example: Solve the equation 3413 =−−+ xx
Identification/Analysis This is an equation (contains an equal sign). It contains two radical terms. Use the steps for equations that contain two radicals.
Solution
( )
( )
( )( )
22
22
2
2
3 1 4 3
3 1 3 4
3 1 3 4
3 1 9 6 4 4
2 4 6 4
2( 2) 6 4
( 2) 3 4
( 2) 3 4
4 4 9( 4)13 40 05 8 0
x x
x x
x x
x x x
x x
x x
x x
x x
x x xx xx x
+ − − =
+ = + −
+ = + −
+ = + − + −
− = −
− = −
− = −
− = −
− + = −
− + =
− − =
0)5( =−x or 0)8( =−x 5=x or 8=x
Isolate one radical by adding 4−x to both sides. Square both sides.
Note: ( )2 223 4 3 4x x+ − ≠ + −
Now there is one radical term left in the equation. Combine like terms and isolate the radical. Factor two out of the terms on the right side. Divide both sides of the equation by a common factor of 2. Squared both sides. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible solutions.
Page 22 of 22
Check Check 5x = : ( )3 5 1 5 4 3+ − − =
15 1 1 3
16 1 34 1 3
3 3
+ − =
− =− =
=
Check 8x = : ( )3 8 1 8 4 3+ − − =
24 1 4 3
25 4 35 2 3
− =
3 3
===
Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Both 5x = and 8x = check, so they are both solutions.
−−
+