- Evaluate the following: √(-25) + √(64) added to √(81) + √(-9)
- Evaluate the following: √(-25) + √(64) divide by √(81) + √(-9)
- Evaluate the following: √(-25) + √(64) minus √(81) + √(-9)
- Evaluate the following: √(-25) + √(64) multiply by √(81) + √(-9)
- Evaluate the following: j^6 + j^7 added to j^3 + j^4
Thursday, March 5, 2009
complex numbers questions 2
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Q1. multiply the root of -1 by the root of 25 and you will get the root of 25j(you will get a positive value as j=the root of -1)and you get 5j and then find the root of 64 which is =8.So the ans for this part is 5j+8.
ReplyDeleteFor the the next part of the question follow the same concept as the first part and you will get 3j+9.
The overal answer for this question is will be
5j+8
+
3j+9 = 8j+17
Q2. tapolin already worked out the first two parts of the problem so the final part is:
ReplyDelete(5j+8)/(3j+9)
to divide complex numbers, you must multiply by the conjugate.
the question now is:
[(5j+8)/(3j+9)]*[(3j-9)/(3j-9)]=
(15j^2+24j-45j-72)/(9j^2+27j-27j-81)=
(15j^2-21j-72)/(9j^2-81)=
(-15-72-21j)/(-9-81)=
(-87-21j)/(-90)
5j+8
ReplyDelete-
3j+9
which equals 2j-1
(√-25 + √ 64) + (√ 81 + √ -9)
ReplyDeleteRecall that √ (-8) = √ -1× √ 8
This implies: √ -1× √25 +√ 64 = 5j + 8
+
√ -1× √9 + √ 81= 3j + 9
Ans = 8j + 17
(j^6 + j^7) + ( j^3 + j^4)
ReplyDeleteRecall j^2= -1
So
j^6 = j^2 × j^2 × j^2
= -1 × -1 × -1
= -1
j^7 = j^2 × j^2 × j^2 × j
= -1× -1 ×-1 × √-1
= -j
j^3 = j^2× j
= -1 × j
= -j
j^4 = j^2 × j^2
= -1 × -1
= 1
this implies that -1 +(-j) +(-j) + 1 = -2j
(√ (-25) + √64) ÷ (√81 + √ (-9))
ReplyDeleteThis Implies:
√ (-25) + √ 64
√ 81 + √ (-9)
In order to solve this we must first make the denominator a real number by multiplying it by its conjugate. The conjugate is the same the denominator but with opposite signs. (The conjugate must be equal to one when simplified) Therefore in this case the conjugate is
√ 81 - √ (-9)
√ 81 - √ (-9)
Therefore this is what we should have:
√ (-25) + √ 64 × √ 81 - √ (-9)
√ 81 + √ (-9) √ 81 - √ (-9)
It can be seen that the conjugate is equal to one
Now we can apply the difference of two squares
(a2 + b2) = (a + b) (a – b)
Calculation for Denominator:
Applying difference of two squares:
This implies:
(√ 81)2 – (√ -9)2
Recall √-9 = √-1 ×√9 = 3j
This implies
81- 3j^2
j^2 = -1
Therefore 81-(-3) = 81+3= 84
Calculation for Numerator
(√ (-25) + √64) (√ 81+ √ (-9))
(√ (-25) × √81) + (√ (-25) × √(-9)) + (√64 × √81) + (√64× √(-9))
Now Simplify
(5j × 9) + (5j × 3j) + (8 × 9) + (8 × 3j)
45j + 15j^2 + 72 + 24j
Recall j^2 = -1
Then Group and solve:
(72 – 15) + (45j + 24j) = 57 + 69j
Overall Solution
Answer: 57 + 69j
84
please note that the ans to the previous ques is (57 + 69j) ÷ 84, sorry the line didn’t come out. ( same with the conjugate)
ReplyDeleteQ3.√(-25) + √(64)-√(81) + √(-9)
ReplyDelete√(-25) + √(64)
√-1√25 + 8
=5j+8
√(81) + √(-9)
=9+√-1√9
=9+3j
Therefore: 8+5j
-9+3j
Ans =-1+2j
hey correct me if im wrong but isnt tapolin wrong about the root of -25??
ReplyDeletethe root of -25=root of 25 * root of -1
therefore it would be root 25j as the final answer?doesnt the root apply over the entire 25j or just the 25?
in which case is it right to find the root of j?
Yes taplin is correct about root 25 being 5.
ReplyDelete(root 25)* (root-1)
(the root of 25 is 5) and the (root of -1 is j)
so simply the answer is 5j. do not get mistaken, (root(25*-1)) is not the same as (root25 * root -1)
hey samos u hav 2find d square root of the 25 first since it is a whole no.Understand??
ReplyDeleteaye jade and ridiclyric the no. 3 u did am not gettin ne of those.This is wat am getting:
ReplyDelete√ (-25) + √ (64) divide by √ (81) + √ (-9)
From this u can get:
√-1 √25 and the √ (64) is 8
Since the √-1 is j and wel the √25 is 5
This nw reads : 5j+8
√ (81) + √ (-9)
The √ (81) is 9 and the √ (-9 can be broken
Up into two pieces, √-1 √9.
The √9 is 3 and wel the √-1 is j
NOW: 5j + 8/3j + 9 X 3j - 9/3j – 9
The numerator
( 5j +8) (3j- 9)
15j2 – 45 + 24j – 72
15j2-21j-72
15(-1)- 21j – 72
-15 – 21j-72
-15-72-21j
-87-21j
The denominator
3j – 9
(3j) – (9) 2
9j2 + 81
9(-1) + 81
-9+ 81
72
1/72 ( 87 – 21j)
desi girl question 3 is not devide its minus, its question 2 you have to devide for
ReplyDeletewhat you did was question 2
ReplyDeletedesi girl i kind ah lost when i reach up to the part (NOW: 5j + 8/3j + 9 X 3j - 9/3j – 9) how you get that.
ReplyDeletewel angel i worked each part at a time and then brought them 2gether...
ReplyDeletefor Q2 why can't the answer be worked like this:
ReplyDeletesq.root-25 + sq.root 64 - sq.root81 + sq.root-9
which will give:
5j + 8 - 9 + 3j
5j + 3j - 9 + 8
= 8j -1
instead of:
5j + 8
-3j -9
=2j - 1
well desi girl how i see it is the denominator will be the same as the differenc of two squares since they are the same but one has a minus and one has a plus.So i applied the equation a^2 - b^2 = ( a + b ) ( a - b). Then i recalled that the root of a number squared is equal to the number. that is how i came up with that answer. It may be kinda confusing because some of the division and brackets did not copy. Please let me kno if you understood how i came to my answer....in ques 2
ReplyDeletei dont understand what ridiclyric did for Q2.
ReplyDeleteisn't the answer supposed to be worked out like this?:
the numerator sq.root-25+sq.root64 divided by the denominator sq.root81+sq.root-9
TAKING NUMERATOR
=>sq. root -25 + sq.root 64
5j + 8
TAKING DENOMINATOR
=>sq.root 81 + sq.root -9
9 + 3j
THEREFORE:
(5j + 8/3j + 9) x (3j - 9/3j - 9(conjugate))
SO NEW NUMERATOR = (5j + 8)x(3j - 9)
NEW DENOMINATOR = (3j + 9)x(3j - 9)
taking denominator:
(3j+9)x(3j-9)
9j^2-27j+27j-81
9(-1)-81
-9-81
=-90
taking numerator:
(5j+8)x(3j-9)
15j^2-45j+24j-72
15(-1)-21j-72
-15-72-21j
-87-21j
TOTAL RESULT:
N/D: -87-21j/90
why do we need to use the denominator particulary to to find the conjugate?
ReplyDeletewhy cant we use the numerator instead?
Hey guys I realise dat you all don’t understand how I got my answer but I will be pleased if you showed me exactly where I went wrong instead of just sayin you don’t know what I did. However I do see one error in the calculation of my denominator with (√-9) ^2 this is suppose to work out to (√-1)2 (√ 9)^2 = (j^2) (3)^2
ReplyDelete= 9j
so it is 81 – 9j^2 = 81 – (-9)
81+9= 90
Therefore my overall solution should be (57+69j) ÷ 90
Hey I found my problem in my calculation for the numerator I mixed up the signs, so instead of multiplying by the conjugate I multiplied by the original equation. Thanks next time I’ll try to do it a lil slower before I make that stupid error again……thanks
ReplyDeleteThis comment has been removed by the author.
ReplyDeleteno 1
ReplyDeletethe first thing to do is to work out anything that can be worked out! dont trail anything, its makes the problem seem difficult
= (-1x25)^1/2 + 8 + 9 + (-1x9)^1/2
=5j + 17 +3j
17 + 8j
no 2
ReplyDeletethe key to this problem is again to work out anything that can be worked out!!! then multiply by the conjugate
= [(8+5j)/(9+3j)] x [(9-3j)(9-3j)]
= (72-24j+45j-15j^2)/(81-9j^2)
=[72-21j-15(-1)]/[81-9(-1)]
= (87 - 21j)/90
no 3
ReplyDeletework out what can be worked out then take away the terms that are in common. pay special attention to opperation signs!!!
=(8+5j) - ((9+3j)
=(-1 + 2j)
no 4
ReplyDeletework out what can be worked out by removing the "shed" ie the square root sign.
= (8+5j) x (9+3j)
multiply each term in the first bracket by each term in the second bracket.
= 72 + 24j + 45j + 15j^2
= 72 + 69j -15
= 57 + 69j
no 5
ReplyDeletej^6 + j^7 + j^3 + j^4 = 1
an easy way to do this is to add all the powers,and since we know that j^2 = -1 we divide the total number of powers by 2. if we get an even number then the answer is 1, but if we get an odd number then the answer is -1.
(Q2) {√ (-25) + √ (64)} ∕ {√ (81) + √ (-9)}
ReplyDeletelet’s call √ (-25) + √ (64) _ X
and √ (81) + √ (-9) _ Y
to divide X by Y , we have to multiply the numerator, X and the denominator, Y by the conjugate of Y.
Two complex numbers of the form a+bj and a-bj are said to be conjugate
Numerator: √ (-25) + √ (64) → (√ -1 * 25) + 8
= j √ 25 + 8
= 5j + 8
Denominator: √ (81) + √ (-9) → 9 + (√ -1 * 9)
= 9 + j √ 9
= 9+ 3j
X/ Y : 5j + 8/9+ 3j____________ the conjugate of Y is 9- 3j
Numerator (5j + 8) (9- 3j) = 87+ 2j
Denominator (9+ 3j) (9- 3j) = 90
Ans. 1/90 (87 +21j)
so the conjugate is simply just the opposite of the denominator. and just multiply the entire equation by its conjugate...? so how come it becomes 1/90 (87-21j) why is it 1/90 and not 90 as found..
ReplyDeleteSantosh i quite don't understand the ending of the question could you probly show all your steps.
ReplyDelete#
ReplyDelete{√(-25) + √(64)} + {√(81) + √(-9)}
{√25√-1 + √(64)} + {√(81) + √9√-1}
(5√-1 + 8) + (9 + 3√-1
using: √-1 = j
5j + 8 + 9 + 3j
adreal to real and imag. to imag.
17 + 8j
# 5
ReplyDelete(j^6 + j^7) + (j^3 + j^4)
using: j^6 = j^2 * j^2* j^2
j^7 = j^2 * j^2* j^2 * j
j^3 = j^2 * j
j^4 = j^2 * j^2
considering: j = √-1
and j^2 = -1
we get:j^6 = -1 * -1 * -1
j^7 = -1 * -1 * -1 * √-1
j^3 = -1 * √-1
j^4 = -1 * -1
and: j^6 = -1
j^7 = -1 * √-1 = -1j
j^3 = -1 * √-1 = -1j
j^4 = 1
therfore we get:(j^6 + j^7) + (j^3 + j^4)
= -1 + -1j + -1j +1
= -2j
Question 2
ReplyDeletethe key is to simplify the question:
= [(8+5j)/(9+3j)] x [(9-3j)x(9-3j)]
= (72-24j+45j-15j^2)/(81-9j^2)
=[72-21j-15(-1)]/[81-9(-1)]
= (87 - 21j)/90
Do anyone diasagree with my answer??
Question 3
ReplyDelete√(-25) + √(64) minus √(81) + √(-9)
Firstly work out what can be worked out then take away the terms that are in common.
=(8+5j) - (9+3j)
Real - Real.........imag - imag
=(-1 + 2j)
Evaluate the following: √(-25) + √(64) added to √(81) + √(-9)
ReplyDeletekeepin in mind that
√(-25) = √(-1) x √(25) = 5j
√(-9) = √(-1) x √(9) = 3j
rewrite and add like terms
(8+5j) + (9+3j)
8 + 5j
+
9 + 3j
_______
17+ 8j
_______
.
Evaluate the following: √(-25) + √(64) minus √(81) + √(-9)
ReplyDeleteto solve this question follow the steps taken in the previous question i did and then minus like terms
8 + 5j
-
9 + 3j
_______
-1+ 2j
_______
Evaluate the following: √(-25) + √(64) divide by √(81) + √(-9)
ReplyDeleteto solve this question follow the steps taken in the previous question i did and then devide
(8 + 5j)÷(9 + 3j)
the same goes for multiplication
ReplyDelete(8 + 5j)x(9 + 3j)