Algebra 1 Glencoe McGraw-Hill JoAnn Evans
Math 8H
10-1Simplifying Radical
Expressions
Simplifying Radical Expressions
The simplest form of a radical expression is an expression that has:
No perfect square factors other than 1 in
the radicand.
not simplified
No fractions in the radicand.
not simplified
No radicals in the denominator of a
fraction.not
simplified
20
21
259
Product Property for Radicals:
The square root of a product equals the product of the
square roots of the factors.
ab a b
32 16 2 42 6 7
Check the validity of these statements with your calculator.
To efficiently simplify radicals using the Product Property, look for the
largest perfect square factor in the radicand.
Any perfect square factors?
Simplify.
The factors on the left worked, but took extra step. When the largest perfect square factor (16) was found, the problem was solved more efficiently.
2 5
4 5 20
48 4 4 3 2 2 3 4 3
48 16 3 4 3
Simplify using the Product Property of Radicals:
4 580
98
54
250
84
243
16 5
7 249 2
3 69 6
5 1025 10
2 214 21
9 381 3
Multiply, then simplify the square roots.
3 5
3 15
45
9 5
Use the product property of radicals.
Are there any perfect square factors of 45?What is the positive square root of 9?
Simplified answer
5 2
5 10
50
25 2
10 3
5 60
300
100 3
6 3 2
2 3 3 6
6 18
6 9 2
18 2
48
6 4 24
4 144
4 12
Quotient Property for Radicals:
The square root of a quotient equals the quotient of the
square roots of the numerator and denominator.
Note: b > 0; division by zero is undefined.
Zero unnnder the line is unnndefined.
a ab b
Place the numerator and denominator under separate radical signs, then simplify each.
If possible, write the fraction in lowest terms.
624
14
1
4
12
Simplify using the Quotient Property of Radicals:
325
2049
1824
327
1136
3
25
35
4 5
49
2 5
7
34
3
4
32
19
13
11
36
116
1
9
Simplify using the Quotient Property of Radicals:
2
140
2
516
4
35
5
10
8 5
95
14 10
2
12 10
2 1 10
516
4
516
2
2
235
5
95
25
8
1
1
5
Rationalize the Denominator
No radical signs may be left in the denominator. To simplify an expression that has a radical in the
denominator, multiply by the BIG GIANT ONE. This is algebraically justified because it is equivalent to
multiplying the original fraction by 1. Multiply the numerator and the denominator by the radical found in the denominator.
Simplify the denominator.
1
2
2
2
1 2
2 2
2
4
22
This answer is fully simplified. The denominator has been rationalized. Remember, a radical expression is not simplified if there is a radical in the denominator.
22
Don’t be fooled into thinking you can cancel the 2’s in this problem. The 2 you see in the numerator is the square root
of 2, not 2.
2 1.4142135...means
2 2
Students often wonder, “Can I cancel a number that’s under a radical and a number that’s not under a radical?”
Rationalize the Denominator
3
7
2
3
18
3
7 52 3
7
7
3
3
18
3
3
3
21
49
217
6
9
63
38
9
38
3
8 33
7 5
2 3 35
6
6
6
210
36
2106
Find the area of a rectangle with the given width and length.
2 15
5
area 5 2 15
2 75
2 25 3
2 5 3
210 3 units
Find the length of the leg of the right triangle using the Pythagorean
Theorem.
2 2 2a b c 2 2 2(3) (x) (7)
29 x 49 2x 402x 40
7"
x
3"
x 4 10
x 2 10 inches
Only the positive root will make sense in this context.
Day 2
Using the Conjugate to Rationalize a
Denominator
Simplify Radicals with Variables
Derivation of the Quadratic Formula
there are no perfect square factors other than 1 in the
radicand.
there are no fractions in the radicand.
there are no radicals in the denominator of a fraction.
Remember, a radical expression isn’t simplified unless
(x – 5) (x + 5)
= x2 - 25
Do you remember the Sum and Difference Pattern
you learned when multiplying binomials?
= x2 + 5x – 5x - 25
(x + 7) (x - 7)
= x2 - 49
= x2 - 7x + 7x - 49
When the sum and difference of two terms are multiplied together, the two middle terms are
opposites and will cancel out, leaving the first and last terms.
The remaining terms will be squares.
Sometimes the denominator of a radical expression may have two terms. The denominator can still be
rationalized using its conjugate.
A radical and its conjugate are the sum and difference of the same two terms. Notice the
pattern we saw earlier:
4 3radical
expression
4 3conjugate
(4 3) (4 3)
16 4 3 4 3 9
16 9
16 3 13
The product is a rational
number.
5
5 3
Simplify by rationalizing the denominator.
No radical expression is simplified if there is a radical in the denominator. What is the conjugate of the denominator?5 3
5 3
5 3
5( 5 3)
( 5 3)( 5 3)
5( 5 3)
25 9
5( 5 3)5 3
5( 5 3)2
Don’t distribute in the numerator until the denominator is rationalized.
5 5 5 32
2
4 5
Simplify by rationalizing the denominator.
No radical expression is simplified if there is a radical in the denominator. What is the conjugate of the denominator?
4 5
4 5
4 5
2(4 5)
(4 5)(4 5)
2(4 5)
16 25
2(4 5)16 5
2(4 5)11
8 2 511
Don’t distribute in the numerator until the denominator is rationalized.
8
2 2
Simplify by rationalizing the denominator.
No radical expression is simplified if there is a radical in the denominator. What is the conjugate of the denominator?
2 2
2 2
2 2
8(2 2)
(2 2)(2 2)
8(2 2)
4 4
8(2 2)4 2
8(2 2)2
8 4 2
4
Don’t distribute in the numerator until the denominator is rationalized.
Think of a radical symbol like a jail and the parts of the radicand as prisoners inside.
Some prisoners will spend their life in the radical “jail”; others will be paroled.
To be released from the radical “jail”, certain requirements must be met.
Rule:A radical will only release parts that are raised to a power that matches its index.
Ideas on this slide and the next from “The Complete Idiot’s Guide to Algebra” by W. Michael Kelley,
2004.
2 332x yThe index on a square root is
2.
How many parts of the radicand are raised to the 2nd
power?2 216 2 x y y
42
16 is 42. x has an exponent of 2.
Rewrite y3 as y2 times y.
Release all parts of the radicand that have a power of 2.
4 xy 2y
Why are the x and y in an absolute value symbol?An even-powered root must have a positive answer. See why on the next slide.
2x x
When finding the principal square root of an expression containing variables, be sure that the result is not
negative.
2( 2) 2It may seem that the answer is…
What if x has a value of -2?
22
Substitute -2 for x in the equation.
?
For radical expressions where the exponent of the variable inside the radical is eveneven and the resulting simplified exponent is oddodd, you must use absolute
value to ensure nonnegative results.
2x 3x
x x
5x
4x x
4 2?
x 2x x 2x x
6x3x
Multiply, then simplify the square roots.
22x y 10y
4 340x y Simplify variable powers too.
Simplified answer
2 24 r s t 2t
2 4 532r s t
2 4 416 2 r s t t 2 24 2 r s t t
5 22 y z 14xz
10 556xy z
10 44 14 x y z z 5 22 14 x y z z
Perfect squares
4 24 10 x y y 22 x y 10 y
5 5y y
6y 12
12 12
6y
12
72y
144
36 2y12
6 2y12
2y2
1
2
3n 8
8 8
3n
8
24n
64
4 6n8
2 6n8
6n4
1
4
Use the Quotient Property of Square Roots to Derive the Quadratic Formula!
Start with standard form of a quadratic. Use the Completing the Square method.
2ax bx c 0
Divide each term by a. When completing the square, a must be 1. What’s next?
2 b cx x 0
a a
Subtract from each side.ca
What will complete the square?
2 b cx x
a a
ba
2 22
2 2
b b c bx x
a a4a 4a
2b2a
Half of , squared.
2 22
2 2
b b c bx x
a a4a 4a Factor as the square of
a binomial.
2 2
2
b c bx
2a a 4a
What would be a common denominator for the 2 fractions on the right side of the equation?
2 2
2 2
b 4ac bx
2a 4a 4a
Combine the 2 fractions over the common denominator.
2 2
2
4ac bbx
2a 4a
Take the square root of each side.
2 2
2
b 4acbx
2a 4a
2 2
2
b 4acbx
2a 4a
2
2
b b 4acx
2a 4a
Use the Quotient Property of Square Roots.
2b b 4acx
2a 2a
What’s the square root of 4a2?
Subtract from both sides.
b2a
2b b 4acx
2a 2a
Combine the two fractions over the common denominator.
2b b 4acx
2a
There it is..... the Quadratic Formula!
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