Solve x and y given that they are rational.
Example 1 :
(x+y√2) (2-√2) = 1+√2
Solution :
(x+y√2) (2-√2) = 1+√2
By distribution, we get
2x-√2x+2√2y–2y = 1+√2
By grouping, we get
-√2x + 2√2y + 2x – 2y = 1+√2
(2x–2y) + (-x+2y)√2 = 1+√2
By equating corresponding terms, we get
2x-2y = 1 ----(1)
-x+2y = 1 ----(2)
Add (1) + (2), we get
2x-2y-x+2y = 1+1
x = 2
By applying the value of x in (2), we get
-2+2y = 1
2y = 3
y = 3/2
So, x = 2 and y = 3/2
Example 2 :
(2 - 3√2) (x + y√2) = √2
Solution :
(2-3√2) (x+y√2) = √2
By distribution, we get
2x+2√2y-3√2x–6y = √2
By grouping like terms
2x–6y-3√2x+2√2y = √2
(2x-6y) + √2(-3x+2y) = 0+√2
Equate the corresponding terms.
2x–6y = 0 ----(1)
-3x+2y = 1 ----(2)
(1)+3(2)
2x-6y-9x+6y = 0+3
-7x = 0+3
x = -3/7
By applying the value of x in (1), we get
2(-3/7) - 6y = 0
-6/7 = 6y
y = -1/7
So, the value of x is -3/7 and y = -1/7.
Example 3 :
(x+y√2) (3+√2) = 1
Solution :
(x+y√2) (3+√2) = 1
By distribution, we get
3x+√2x+3y√2+2y = 1
By grouping the like terms.
(3x+2y) + (x+3y)√2 = 1 + 0
3x + 2y = 1 -----(1)
x + 3y = 0 -----(2)
(1)-3(2) ==>
3x+2y - 3x-9y = 1-0
-7y = 1
y = -1/7
We get,
Now, y = - 1/7
By applying the value of x in (1)
We get,
x + 3y = 0
x + 3(-1/7) = 0
x – 3/7 = 0
x = 3/7
So, x = 3/7 and y = - 1/7
Find rational a and b such that :
Example 4 :
(a+√2) (2-√2) = 4-b√2
Solution :
(a+√2) (2-√2) = 4-b√2
By distribution, we get
2a-√2a+2√2–2 = 4-b√2
(2a-2)+(2-a)√ 2 = 4-b√2
2a–2 = 4 -----(1)
2-a = -b -----(2)
From (1),
2a = 6
a = 3
By applying, the value of a in (1)
2-3 = -b
-1 = -b
So, a = 3
and b =
- 1
Example 5 :
(a + b√2)2 = 33 + 20√2
Solution :
Given, (a + b√2)2 = 33 + 20√2
By using algebraic identity, we get
(a + b)2 = a2 + 2ab + b2
a2 + 2ab√2 + 2b2 = 33 + 20√2
(a2+2b2)+ 2ab√2 = 33+20√2
a2+2b2 = 33 ----(1)
2ab = 20
ab = 10 ----(2)
By applying the value of b in (1),
We get,
a2 + 2 (10/a)2 = 33
a2 + 2 (100/a2) = 33
a2 + 200/a2 = 33
a4 + 200 = 33a2
a4 -33a2+ 200 = 0
Let t = a2
t2 – 33t +
200 =
0
By factorization, we get
(t-25)(t-8) = 0
t = 25 and t = 8
a2 = 25 and a2 = 8
a = 5, a = 2
Example 6 :
(x+y√2) (3-√2) = -4√2
Solution :
By distribution, we get
3x-√2x+3y√2–2y = -4√2
(3x-2y)+(-x+3y)√2 = 0-4√2
Equating corresponding terms, we get
3x–2y = 0 -----(1)
-x+3y = -4 -----(2)
(1) + 3(2)
3x-2y-3x+9y = 0-12
7y = -12
y = -12/7
By applying the value of y in (1), we get
3x-2(-12/7) = 0
3x+24/7 = 0
3x = -24/7
x = -8/7
So, the value of x is -8/7 and y is -12/7
Example 7 :
Find √[(x2/36) + (x2/25)]
Solution :
= √[(x2/36) + (x2/25)]
= √[(25x2 + 36x2)/25(36)]
= √[(61x2/(5⋅5⋅6⋅6)]
= √(61x2)/(5⋅6)]
= (x/30)√61
= (x√61/30)
So, the answer is (x√61/30)
Example 8 :
If 2/x = √0.16, then x equal.
Solution :
2/x = √0.16
2/x = √0.16 x (100/100)
2/x = √(16/100)
2/x = √(4⋅4/10⋅10)
2/x = 4/10
Doing cross multiplication, we get
20 = 4x
x = 20/4
x = 5
So, the value of x is 5.
Example 9 :
Simplify √[(y2/2) - (y2/18)]
Solution :
= √[(y2/2) - (y2/18)]
= √[(18y2 - 2y2)/2(18)]
= √[16y2/2(18)]
= √[4⋅4⋅y⋅y/2⋅2⋅3⋅3]
= 4y/6
Simplifying the fraction, we get
= 2y/3
Example 10 :
Simplify √√x = xa, what is the value of a ?
Solution :
√√x = xa
Raising power 2 on both sides
(√√x)2 = (xa)2
√x = x2a
Raising power 2 on both sides
(√x)2 = (x2a)2
x = x4a
Since bases are equal on both sides, we can equate the powers
1 = 4a
a = 1/4
So, the value of a is 1/4
Example 11 :
2√(x + 2) = 3√2
If x > 0 in the equation above, what is the value of x ?
a) 2.5 b) 3 c) 3.5 d) 4
Solution :
2√(x + 2) = 3√2
Dividing by 2 on both sides
√(x + 2) = 3√2/2
Raising power 2 on both sides, we get
[√(x + 2)]2 = (3√2/2)2
x + 2 = 9(2)/4
x + 2 = 9/2
Subtracting 2 on both sides
x = (9/2) - 2
x = (9 - 4)/2
x = 5/2
x = 2.5
So, the value of x is 2.5.
Example 12 :
Which of the following is equivalent to x2a/b, for all values of x ?
Solution :
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