rendered paste bodyROOTS AND RADICALS II
NOTES:
Before pressing ENTER, describe in your own words how we multiply radicals.
_ _
\/a * \/b (a >= 0 & b >= 0)
First we multiply the radicands and then simplify.
_ _ __
\/a * \/b = \/ab
_ _
So far the symbols \/a and -\/a have been used to stand for solutions to
2
x = a
Many textbooks, tests, and courses also use another notation.
1/2 1/2
The symbols a and -a often are used to stand for the solution to
2
x = a
1/2 _
That is a and \/a are different symbols standing for the same thing.
1/2 _
Because a = \/a we know that:
1/2 2 1/2 1/2
(a ) = (a ) * (a )
_ _
\/a * \/a = a
Here is why the notation:
1/2
a
makes sense as a solution to the equation:
2
x = a
From the unit on polynomials, we know that:
m n m+n
x * x = x
So, in particular, it should be true that:
1/2 1/2 1/2+1/2 1
x * x = x or x or x
This is what we wanted!
***
Notice how we divide radicals.
__ __
/25 \/25 5
/ -- = ---- = -
\/ 9 \/9 3
***
__ __ _____
/72 \/72 \/9*4*2
/ -- = ---- = -------
\/ 50 \/50 \/25*2
_ _
3*2\/2 6\/2 6
------ = ---- = -
5\/2 5\/2 5
Here is another way to do the same problem:
__ __ __
/72 /36 \/36 6
/ -- = / -- = ---- = -
\/ 50 \/ 25 \/25 5
*Is that enough? Good, I'm tired of making ASCII art for it. :-P *
***
Notice what we do to simplify a fraction when a radical appears in the denominator.
_ _
3 3 \/3 3*\/3
---- = ---- * --- = ---------
2\/3 2\/3 \/3 2*\/3*\/3
_ _
3\/3 \/3
---- = ---
2*3 2
*Get it? Good. There are too many more of these, hehe! ;-) *
This process is callled rationalizing the denominator and usually is done when a radical appears in the denominator.
***
Before pressing ENTER, desribe in your own words how we divide radicals.
_
/a
/ - (a >= 0 & b > 0)
\/ b
Reduce the fraction a/b then rewrite the quotient as one radical divided by another and simplify. If a radical remains in the denominator, rationalize the denominator.
_ _
/a \/a
/ - = ---
\/ b \/b
Notice how we simplify expressions containing radicals:
_ _ _ _
3\/7 + 8\/7 = (3 + 8)\/7 = 11\/7
***
_ _ _ _
5\/6 - 12\/6 = (5 - 12)\/6 = -6\/6
***
__ __ __
9\/3x + 12\/3x - 18\/3x
__ __
(9 + 12 - 18)\/3x = 3\/3x
***
Sometimes we must simplify individual terms in a radical expression BEFORE adding and subtracting.
__ __ _ _ _ _
3\/12 + 5\/27 = 3\/4 \/3 + 5\/9 \/3
_ _
3 * 2\/3 + 5 * 3\/3
_ _
6\/3 + 15\/3
_
21\/3
*GAH! Too many radicals. :-o ... ;-) *
Before pressing ENTER, desribe in your own words how we add and subtract expressions containing square roots.
Write each term in the expression in simplest form and then combine terms with common factors.
WORKSPACE:
__ __ _________________
\/21 * \/42 = \/2 * 3 * 3 * 7 * 7
_
21\/2
***
_ ___ _____________
3\/5 * \/125 = 3\/5 * 5 * 5 * 5
75
***
___ ___
-2\/15t * 3\/45t
***
__ __
\/28 * (-\/42)
_____________________
-\/2 * 2 * 2 * 3 * 7 * 7
_
-14\/6
***
___ ___ _
6\/12k * \/18k * \/6
_____________________
6\/4 * 4 * 9 * 9 * k * k
216k
***
__ _ __
3\/27 - 2\/3 + 6\/12
___ _ ___
3\/3*9 - 2\/3 + 6\/3*4
_ _ _
9\/3 - 2\/3 + 12\/3
19\/3
***
_ __ __ __
9\/5 - 7\/20 + 4\/45 - \/10
_ ___ ___ __
9\/5 - 7\/5*4 + 4\/5*9 - \/10
_ _ _ __
9\/5 - 14\/5 + 12\/5 - \/10
_ __
7\/5 - \/10
***
__ ___ __
5t\/9t + 6t\/12t - 4t\/4t
_ __ _
15t\/t + 12t\/3t - 8t\/t
_ __
7t\/t + 12t\/3t
***
_ _ __
(2\/5)(-6\/3)(\/15)
_____________
-12\/3 * 3 * 5 * 5
-180
***
_ __ _
2\/7 + 3\/28 - 4\/7
_ _________ _
2\/7 + 3\/2 * 2 * 7 - 4\/7
_ _ _
2\/7 + 6\/7 - 4\/7
_
4\/7
***
_
6\/2
----
9\/3
*Aha! Multiply the denominator radical by both the numerator and denominator*
***
_ _
-6\/3 - (-6\/3)
0
***
__ __
/25 \/25 5
/ -- = ---- = -
\/ 4 \/4 2
***
__ __ __
2\/50 + 4\/72 - 5\/98
_ _ _
10\/2 + 24\/2 - 35\/2
_
WRONG: -1\/2
_
RIGHT: -\/2
*Note: Do NOT add a 1 in front of a radical!! Remove it first!*
***
_____ _
/3 6 /9
/ - * - = / - = 3/2
\/ 4 2 \/ 4
***
__ ___ ___
4\/3x + 5\/12x - 6\/27x
__ __ __
4\/3x + 10\/3x - 18\/3x
__
-4\/3x
***
_ __ __ _
(3\/3)(2\/10)(-6\/15)(\/8)
____
-36\/3600
-2160
***
ANS:
21\/2 , 75 , -90t\/3 , -14\/6 , 216k , 2 , 2/3 , 7/9 , 12/11 , 1/2 , \/7 \/7 , 5*\/7 , 7 , 5\/7 14 , \/6 \/6 , 6*\/6 , 6 , 3 , 7\/5 15 , -\/6 15 , 3\/15 4 , 9\/2 7 , -2\/21 33 , 23\/3 , 0 , 7\/x , 4\/23 , -7\/11 , 5p\/3 , 26\/x , 23\/6 , 38\/2 , 19\/3 , 7\/5 - \/10 , -10\/6 - 40\/2 , -\/6 , -180 , 4\/7 , 0 , 5/2 , -\/2 , 3/2, -4\/3x , -2160
DATE: February 16, 2006. The above was continued.
ROOTS AND RADICALS: Simplifying Radical Expressions
- Multiply binomial expressions containing radicals
- Rationalize denominators containing binomials with radicals
NOTES:
Remember:
2 2
(a + b)(a - b) = a - b
2 2 2
(a + b) = a + 2ab + b
2 2 2
(a - b) = a - 2ab + b
We can use these products to simplify binomial products involving radicals.
Let's look at a product of the form:
2 2
(a + b)(a - b) = a - b
_
For now, let a = 3 and b = \/5.
_ _ 2 _ 2
(3 + \/5)(3 - \/5) = 3 - (\/5)
_
Or, let a = x and b = 2\/3.
_ _ 2 2
(x - 2\/3)(x + 2\/3) = x - (2\/3)
2 2 _ 2 2
x - 2 (\/3) = x - 12
***
Let's look at products of the form:
2 2 2
(a + b) = a + 2ab + b
_
For now, let a = 6 and b = 2\/3.
_ 2 2 _ _ 2
(6 + 2\/3) = (6) + 2(6)(2\/3) + (2\/3)
_
36 + 24\/3 + 12
_
48 + 24\/3
_
Or, let a = x and b = 3\/7.
_ 2 2 _ _ 2
(x + 3\/7) = (x) + 2(x)(3\/7) + (3\/7)
2 __
x + 6\/7x + 63
***
Finally, let's look at products of the form:
2 2 2
(a - b) = a - 2ab + b
_
For now, let a = 4 and b = \/5.
_ 2 2 _ _ 2
(4 - \/5) = 4 - 2(4)(\/5) + (\/5)
_
16 - 8\/5 + 5
_
21 - 8\/5
_
Or, let a = x and b = \/7.
_ 2 2 _ _ 2
(x - \/7) = x - 2(x)(\/7) + (\/7)
2 __
x - 2\/7x + 7
WORKSPACE:
_ __ _ __
(\/6 - \/10)(\/6 + \/10)
__ __ __ ___
\/36 + \/60 - \/60 - \/100
6 - 10 = -4
***
_ _
(t - 3\/3)(t + 3\/3)
2 _ _ _
t + 3t\/3 - 3t\/3 - 9\/3
2
t - 27
*Note: Remember to multiply EVERYTHING ;-) *
***
_ _
(2 - \/7)(2 + \/7)
_ _ __
4 + 2\/7 - 2\/7 - \/49
4 - 7 = -3
***
_ _
(k + 5\/3)(k - 5\/3)
2 _ _ _
k - 5k\/3 + 5k\/3 - 25\/9
2
k - 75
***
_ 2
(2 + 5\/2)
*Hmmm... I don't get it, so here is everything from the screen :-) *
_
ANSWER: 54 + 20\/2
2 2 2
(a + b) = a + 2ab + b
_ 2
(2 + 5\/2)
2 _ _ 2
(2) + 2(2)(5\/2) + (5\/2)
_
ANSWER: 54 + 20\/2
*This does not seem right... where does that second 2 come in? :-| *
***
*EEK! Not another one...*
_ 2
(-4 + 6\/5)
__
16 + 36\/25
*196? Nope.*
_
16 + 36\/5
*Not this either! /me is banging head*
*...well, let's try that info they gave me above :-( *
_
16 + 16 + 12\/5
*WHAT? That's not even right. Bah.*
_
ANSWER: 196 - 48\/5
*Stepping through...*
_ 2
(-4 + 6\/5)
2 2 2
(a + b) = a + 2ab + b
_ 2
(-4 + 6\/5)
2 _ _ 2
(-4) + 2(-4)(6\/5) + (6\/5)
_
ANSWER: 196 - 48\/5
*Hmmm...*
***
_ 2
(2 + 9\/2)
_ _
4 + 2(2)(9\/2) + 81\/4
_ _
4 + 36\/8 + 81\/4
_
4 + 72\/2 + 162
_
166 + 72\/2
*Aaargh!! Let's try it AGAIN (this could take awhile).*
_ 2
(2 + 9\/2)
2 2 2
(a + b) = a + 2ab + b
_ _
4 + 4(9\/2) + 81\/4
_
166 + 36\/2
*Yeees!!! It's finally right!*
***
_ 2
(t + 3\/6)
2 2 2
(a + b) = a + 2ab + b
2 _ __
t + 6t\/6 + 9\/36
2 _
t + 6t\/6 + 54
***
_ 2
(3 - \/2)
2 2 2
(a - b) = a - 2ab + b
_ _
9 - 6\/2 + \/4
_
11 - 6\/2
***
_ 2
(-3 - \/6)
2 2 2
(a - b) = a - 2ab + b
_ __
9 + 6\/6 + \/36
_
15 + 6\/6
***
ANS:
-4 , t^2 - 27 , -3 , k^2 - 75 , 54 + 20\/2 , 196 - 48\/5 , 166 + 36\/2 , t^2 + 6t\/6 + 54 , 11 - 6\/2 , 15 + 6\/6
DATE: February 21, 2006. The above is continued (same lesson).
NOTES:
Can use this format:
_ 2
(x - a\/y)
Is the same thing as:
_ _
(x - a\/y)(x - a\/y)
Conjugates are the "opposite," like:
_ _
2 + \/3 2 - \/3
_ _
-3\/2 + 2 2 + 3\/2
Earlier in this topic we found the product of a radical expression and its conjugate.
_ _ 2 _ 2
(5 - \/7)(5 + \/7) = 5 - (\/7)
25 - 7 = 18
***
Before pressing ENTER, describe in your own words what the conjugate of a binomial radical expression is, and what the product of an expression and its conjugate is.
The conjugate of a binomial radical expression of the form a + b\/c is a - b\/c. Their product is always a rational number.
***
In a previous topic we learned that when there was a radical in the denominator of a fraction, we rationalized the denominator.
Notice how we rationalize the denominator of the following fraction.
_ _
2 2 3+\/5 2(3 + \/5)
----- = ----- * ----- = -2-------2
3-\/5 3-\/5 3+\/5 3 - (\/5)
_ _
6+2\/5 3+\/5
------ = -----
4 2
Here are more examples of rationalizing denominators containing radical expressions.
[one clipped]
_ _
5 5 3-\/6 5(3-\/6)
----- = ----- * ----- = -2-----2
\/6+3 3+\/6 3-\/6 3-(\/6)
_
15 - 5\/6
---------
3
***
Before pressing ENTER, desribe in your own words how to rationalize the denominator of a fraction which has a radical expression in the denominator.
We multiply both the numerator and the denominator of the fraction by the conjugate of the denominator.
***
WORKSPACE:
_ 2
(3 - \/2)
2 2 2
(a - b) = a - 2ab + b
_
9 - 6\/2 + 2
_
11 - 6\/2
***
_ 2
(-3 - \/6)
2 2 2
(a - b) = a - 2ab + b
_
9 - 6\/6 + 6
_
15 + 6\/6
***
_ 2
(5 - 2\/6)
2 2 2
(a - b) = a - 2ab + b
_
WRONG: 25 - 4\/6 + 6
*Hmmm... running through steps given*
_ 2
(5 - 2\/6)
2 _ _ 2
(5) - 2(5)(2\/6) + (2\/6)
*I see, I forgot to multiply EVERYTHING*
_
49 - 20\/6
***
_ _
(4 - 2\/7)(4 + 2\/7)
16 - 28 = -12
***
WRONG:
_ _
9 3+\/3 9(3+\/3)
----- * ----- = --------
3-\/3 3+\/3 9 - 3
*Stepping though...*
RIGHT:
_
9 + 3\/3
--------
2
_
9 3+\/3 9(3+\/3)
----- * ----- = -2-----2
3-\/3 3+\/3 3-(\/3)
_
9 + 3\/3
--------
2
***
_ _ _
8 6-\/7 8(6-\/7) 48-8\/7
----- * ----- = -2-----2 = -------
6+\/7 6-\/7 6-(\/7) 29
***
WRONG:
_ _
7 7 6+2\/5 7(6+2\/5)
------ = ------ * ------ = -2------2
2\/5+6 6-2\/5 6+2\/5 6-(2\/5)
_
42 + 14\/5
----------
16
RIGHT:
_
21 - 7\/5
---------
8
***
WRONG:
_ _
3 3 2-4\/2 3(2-4\/2)
------ = ------ * ------ = -2------2
4\/2-2 2+4\/2 2-4\/2 2-(4\/2)
_
6 - 12\/2
---------
4 - 16\/2
*Got to work this one out...*
_ _
3 3 -2-4\/2 3(-2-4\/2)
------ = ------- * ------- = ----2------2
4\/2-2 -2+4\/2 -2-4\/2 (-2)-(4\/2)
_
3 + 6\/2
--------
14
***
WRONG:
_ _
4 4 1-\/3 4(1-\/3)
----- = ----- * ----- = -2-----2
\/3+1 1+\/3 1-\/3 1-(\/3)
*Must figure this out... *sigh* *
_ _
4 4 1-\/3 4-4\/3
----- = ----- * ----- = ------
\/3+1 1+\/3 1-\/3 1 - 3
_
-2 + 2\/3
*Hmmm... that makes sense. Why didn't they show this to me WAY BEFORE?!*
***
5 3+5\/2 15+25\/2 -15 - 25\/2
------ * ------ = -------- = -----------
3-5\/2 3+5\/2 9 - 50 41
***
_ _
-2 -2 5-\/6 -10+2\/6
----- = ----- * ----- = --------
\/6+5 5+\/6 5-\/6 19
***
ANS:
49 - 10\/6 , x^2 - 8x\/3 + 48 , // 11 - 6\/2 , 15 + 6\/6 , 49 - 20\/6 \\ , 22 , -1 , 19 , -12 , 62 , ... , 27 + 10\/2 , 8 - 2\/7 , -68 , ... , -2 + 2\/3 , 64 + 24\/7 , 86 - 18\/5 , -1 , ... , 102 - 28\/2 , 63 - 18\/6 , 23 , ...
ROOTS AND RADICALS II: Test
This lesson will evaluate your skills in simplifying expressions which contain roots and radicals.
NOTES:
WORKSPACE:
ANS:
d , a , a , c , b , b , d , c , a , b
QUADRATIC EQUATIONS AND THE DISTANCE FORMULA I: Pre-Test
This lesson will evaluate your skills in finding the solutions to quadratic equations and using the distance formula.
*Note to self: I don't know this :-| *
*OK, some of it, I just did one that seemed quite reasonable :-D *
NOTES:
WORKSPACE:
ANS:
NOT:a , b , NOT:a , NOT:c , c , c , NOT:b , NOT:d , NOT:b , a