Number system 1surds & indices

If 32x = 81, what is the value of x?
1
2
3
4
12-Jun-15
C.S.VEERARAGAVAN 98948 34264
veeraa1729@gmail.com 2

34 = 81.
2x = 4
X = 2.
12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 3
If 32x = 81, what is the value of x?

5 + 2
– 5 + 2
Either 1 or 2
None of these.
12-Jun-15
What is the conjugate of 5–2?

1. 5 + 2
2. – 5 + 2
3. Either 1 or 2
4. None of these.
Conjugate is – 5 – 2

2 – 2
2 – 3
3 – 2
2 – 2 2
12-Jun-15

1
3+ 2
=
1
3+ 2
x
3− 2
3− 2

1
3+ 2
=
1
3+ 2
x
3− 2
3− 2
=
3− 2
3−2

1
3+ 2
=
1
3+ 2
x
3− 2
3− 2
=
3− 2
3−2
= 3 − 2

1
3+ 2
=
1
3+ 2
x
3− 2
3− 2
=
3− 2
3−2
= 3 − 2
1
2 + 3
𝑋
2 − 3
2 − 3

1
3+ 2
=
1
3+ 2
x
3− 2
3− 2
=
3− 2
3−2
= 3 − 2
1
2 + 3
𝑋
2 − 3
2 − 3
=
2 − 3
4 − 3

1
3+ 2
=
1
3+ 2
x
3− 2
3− 2
=
3− 2
3−2
= 3 − 2
1
2 + 3
𝑋
2 − 3
2 − 3
=
2 − 3
4 − 3
=2− 3

1
3+ 2
=
1
3+ 2
x
3− 2
3− 2
=
3− 2
3−2
= 3 − 2
1
2 + 3
𝑋
2 − 3
2 − 3
=
2 − 3
4 − 3
=2− 3
Adding we get 2 – 2

What is the positive square root of 48 − 27?
3
1
2
3
1
8
3
1
4
None of these

48 − 27 = 16(3) − 9(3)

48 − 27 = 16(3) − 9(3)
4 3 − 3 3

48 − 27 = 16(3) − 9(3)
4 3 − 3 3
3

What is the positive square root of 7 + 4 3
3+1
2 + 2 3
2 + 3
2 + 3

3+1
2 + 2 3
2 + 3
2 + 3
7 + 4 3 = 7 + 2 3(4)

3+1
2 + 2 3
2 + 3
2 + 3
7 + 4 3 = 7 + 2 3(4)
3 + 4 + 2 3(4) = 3 + 4
2

What is the positive square root of 72 − 40
2 5−1
4
2 5−1
4
2 3 − 2
4
2 5 − 3

2 5−1
4
2 5−1
4
2 3 − 2
4
2 5 − 3
72 − 40

2 5−1
4
2 5−1
4
2 3 − 2
4
2 5 − 3
72 − 40
36(2) − 20(2)

2 5−1
4
2 5−1
4
2 3 − 2
4
2 5 − 3
72 − 40
36(2) − 20(2)
2 36 − 20

2 5−1
4
2 5−1
4
2 3 − 2
4
2 5 − 3
72 − 40
36(2) − 20(2)
2 36 − 20
2 6 − 2 5

2 5−1
4
2 5−1
4
2 3 − 2
4
2 5 − 3
72 − 40
36(2) − 20(2)
2 36 − 20
2 6 − 2 5
2 5 + 1 − 2 5

5
3
2
3−6
1
3+2
2
3
=
31 3
+ 21 3
31 3
− 21 3
21 3
− 31 3
32 3
+ 22 3

5
3
2
3−6
1
3+2
2
3
=
31 3
+ 21 3
31 3
− 21 3
21 3
− 31 3
32 3
+ 22 3
5
3
2
3−6
1
3+2
2
3
5
3
1
3
2
−(2)
1
3(3)
1
3+ 2
1
3
2

5
3
2
3−6
1
3+2
2
3
=
31 3
+ 21 3
31 3
− 21 3
21 3
− 31 3
32 3
+ 22 3
5
3
2
3−6
1
3+2
2
3
5
3
1
3
2
−(2)
1
3(3)
1
3+ 2
1
3
2
5
3
1
3
2
−(2)
1
3(3)
1
3+ 2
1
3
2 𝑥
3
1
3+2
1
3
3
1
3+2
1
3

5
3
2
3−6
1
3+2
2
3
=
32 3
+ 21 3
31 3
− 21 3
21 3
− 31 3
32 3
+ 22 3
5
3
2
3−6
1
3+2
2
3
5
3
1
3
2
−(2)
1
3(3)
1
3+ 2
1
3
2
5
3
1
3
2
−(2)
1
3(3)
1
3+ 2
1
3
2 𝑥
3
1
3+2
1
3
3
1
3+2
1
3
5 3
1
3+2
1
3
3+2

What is the rationalizing factor of 41 3
− 31 3
4
2
3 + 12 + 3
2
3
4
2
3 − 12 + 3
2
3
4
2
3 + 12
1
3 + 3
2
3
4
4
3 + 12
2
3 + 3
4
3

− 31 3
4
2
3 + 12 + 3
2
3
4
2
3 − 12 + 3
2
3
4
2
3 + 12
1
3 + 3
2
3
4
4
3 + 12
2
3 + 3
4
3
41 3
− 31 3
is of the form a – b

− 31 3
4
2
3 + 12 + 3
2
3
4
2
3 − 12 + 3
2
3
4
2
3 + 12
1
3 + 3
2
3
4
4
3 + 12
2
3 + 3
4
3
41 3
− 31 3
is of the form a – b
a2+ab+ b2 should be multiplied
to get a3−b3
4
2
3 + 4
1
33
1
3 + 3
2
3

Simplify: x
a
a+b+c.x
b
a+b+c. x
c
a+b+c
1
x
0
xabc

Which is greater?
1
8+ 5
,
1
2+ 5
1
8 + 5
1
2 + 5
Cannot be determined
None of these.

Which is greater?
1
8+ 5
,
1
5+ 2
1
8+ 5
1
5+ 2
None of these.
8 > 5

Which is greater?
1
8+ 5
,
1
5+ 2
1
8+ 5
1
5+ 2
None of these.
8 > 5 5 > 2

Which is greater?
1
8+ 5
,
1
5+ 2
1
8+ 5
1
5+ 2
None of these.
8 > 5 5 > 2
8 + 5 > 5 + 2

Which is greater? 222
, 322
222
322
Both are equal

–6 – 3 5
5 − 2
– 5 − 2
None of these
What is the rationalizing factor of 6 + 3 5

–6 – 3 5
5 − 2
– 5 − 2
None of these
6 + 3 5

–6 – 3 5
5 − 2
– 5 − 2
None of these
6 + 3 5 3(2) + 3 5

–6 – 3 5
5 − 2
– 5 − 2
None of these
6 + 3 5 3(2) + 3 5
3(2 + 5)

–6 – 3 5
5 − 2
– 5 − 2
None of these
6 + 3 5 3(2) + 3 5
3(2 + 5)
5 − 2 is the factor req.

–6 – 3 5
5 − 2
– 5 − 2
None of these
6 + 3 5 3(2) + 3 5
3(2 + 5)

What is the value of x if 2 𝑥 𝑥
= 16
0
1
3
2

If 2 𝑥
3 𝑦
= 216 and x and y are integers then x,y?
3,2
2,2
3,4
3,3

If 3
1
23
3
23
5
23
7
2 = 9 𝑥
, what is the value of x?
4
3
2
1

(17)3.5 x (17)? = 178
2.29
2.75
4.25
4.5

If
𝑎
𝑏
𝑥−1
=
𝑏
𝑎
𝑥−3
,then the value of x is
2
1
1
2
7
2

If
𝑎
𝑏
𝑥−1
=
𝑏
𝑎
𝑥−3
2
1
1
2
7
2
𝑎
𝑏
𝑥−1
=
𝑏
𝑎
𝑥−3
𝑎
𝑏
𝑥−1
=
𝑎
𝑏
−(𝑥−3)

If
𝑎
𝑏
𝑥−1
=
𝑏
𝑎
𝑥−3
2
1
1
2
7
2
𝑎
𝑏
𝑥−1
=
𝑏
𝑎
𝑥−3
𝑎
𝑏
𝑥−1
=
𝑎
𝑏
−(𝑥−3)
x – 1 = 3 – x
Solving x = 2

Given 100.48 = x, 100.7 = y and xz = y2, value of z = ?
1.45
1.88
2.9
3.7

1.45
1.88
2.9
3.7
xz = y2
100.48z = 102*0.7

1.45
1.88
2.9
3.7
xz = y2
100.48z = 102*0.7
0.48z = 1.4

1.45
1.88
2.9
3.7
xz = y2
100.48z = 102*0.7
0.48z = 1.4
Z =
140
48
=
35
12
≈ 2.9

If 5a = 3125, then the value of 5a – 3 is?
25
125
625
1625

If 3x – y = 27 and 3x + y = 243, then x =?
0
2
4
6

If 3x – y = 27 and 3x + y = 243, then x = ?
0
2
4
6
3x – y = 27 = 33
3x + y = 243 = 35

If 3x – y = 27 and 3x + y = 243, then x = ?
0
2
4
6
3x – y = 27 = 33
3x + y = 243 = 35
x – y = 3

If 3x – y = 27 and 3x + y = 243, then x = ?
0
2
4
6
3x – y = 27 = 33
3x + y = 243 = 35
x – y = 3
x + y = 5

If 3x – y = 27 and 3x + y = 243, then x = ?
0
2
4
6
3x – y = 27 = 33
3x + y = 243 = 35
x – y = 3
x + y = 5
Solving x = 4

2560.16 x 2560.09 = ?
4
16
64
256.25

2560.16 x 2560.09 = ?
4
16
64
256.25
2560.16 x 2560.09 = 2560.25

2560.16 x 2560.09 = ?
4
16
64
256.25
2560.16 x 2560.09 = 2560.25
256 = 44

1
1+x b−a +x c−a
+
1
1+x a−b +x c−b
+
1
1+x b−c +x a−c
= ?
0
1
Xa – b – c
None of these

1
1+x b−a +x c−a
+
1
1+x a−b +x c−b
+
1
1+x b−c +x a−c
= ?
0
1
Xa – b – c
None of these
1
1+
xb
xa +
xc
xa
+
1
1+
xa
xb
+
xc
xb
+
1
1+
xb
xc +
xa
xc

1
1+x b−a +x c−a
+
1
1+x a−b +x c−b
+
1
1+x b−c +x a−c
= ?
0
1
Xa – b – c
None of these
1
1+
xb
xa +
xc
xa
+
1
1+
xa
xb
+
xc
xb
+
1
1+
xb
xc +
xa
xc
xa
xa+xb+xc
+
xb
xa+xb+xc
+
xc
xa+xb+xc

257.5 x 52.5 ÷1251.5 = 5?
8.5
13
16
17.5

243
n
5 X 32n+1
9nX3n−1
1
2
9
3n

243
n
5 X 32n+1
9nX3n−1
1
2
9
3n
35
n
5 X32n+1
32 n
X 3n−1

243
n
5 X 32n+1
9nX3n−1
1
2
9
3n
35
n
5 X32n+1
32 n
X 3n−1
=
3 𝑛+2𝑛+1
32𝑛+𝑛−1

243
n
5 X 32n+1
9nX3n−1
1
2
9
3n
35
n
5 X32n+1
32
n
X 3n−1
=
3n+2n+1
32n+n−1 = 33n + 1 – 3n + 1

1
1+a n−m +
1
1+a m−n = ?
0
1
2
1
am+n

1
1+a n−m +
1
1+a m−n = ?
0
1
2
1
am+n
1
1+an
am
+
1
1+am
an
=

1
1+a n−m +
1
1+a m−n = ?
0
1
2
1
am+n
1
1+an
am
+
1
1+am
an
=
𝑎 𝑚
𝑎 𝑚+𝑎 𝑛 +
𝑎 𝑛
𝑎 𝑚+𝑎 𝑛

If mn = 121, the value of (m – 1)n+1 is
1
10
121
1000

xb
xc
b + c − a
xc
xa
c +a −b
xa
xb
a +b −c
xabc
1
xab+bc+ca
xa+b+c

xb
xc
b + c − a
xc
xa
c +a −b
xa
xb
a +b −c
xabc
1
xab+bc+ca
xa+b+c
xb
xc
b + c − a
xc
xa
c +a −b
xa
xb
a +b −c

xb
xc
b + c − a
xc
xa
c +a −b
xa
xb
a +b −c
xabc
1
xab+bc+ca
xa+b+c
xb
xc
b + c − a
xc
xa
c +a −b
xa
xb
a +b −c
= x(b - c)(b + c - a) . x(c - a)(c + a - b) . x(a - b)(a + b - c)

xb
xc
b + c − a
xc
xa
c +a −b
xa
xb
a +b −c
xabc
1
xab+bc+ca
xa+b+c
xb
xc
b + c − a
xc
xa
c +a −b
xa
xb
a +b −c
= x(b - c)(b + c) – (b-c)(a) . x(c - a)(c + a) – (c-a)(b) . x(a - b)(a + b – (a-b)(c)

xb
xc
b + c − a
xc
xa
c +a −b
xa
xb
a +b −c
xabc
1
xab+bc+ca
xa+b+c
xb
xc
b + c − a
xc
xa
c +a −b
xa
xb
a +b −c
= x(b - c)(b + c) – (b-c)(a) . x(c - a)(c + a) – (c-a)(b) . x(a - b)(a + b – (a-b)(c)
=𝑥 𝑏2−𝑐2+𝑐2−𝑎2+𝑎2−𝑏2
∗ 𝑥 𝑏𝑎−𝑐𝑎+𝑏𝑐−𝑎𝑏+𝑎𝑐−𝑏𝑐

If x = 3 + 2 2, then 𝑥 −
1
𝑥
is
1
2
2 2
3 3

If x = 3 + 2 2, then 𝑥 −
1
𝑥
is
1
2
2 2
3 3
X = 3 + 2 2

If x = 3 + 2 2, then 𝑥 −
1
𝑥
is
1
2
2 2
3 3
X = 3 + 2 2
𝑥 −
1
𝑥
2
= 𝑥 +
1
𝑥
− 2

If x = 3 + 2 2, then 𝑥 −
1
𝑥
is
1
2
2 2
3 3
X = 3 + 2 2
𝑥 −
1
𝑥
2
= 𝑥 +
1
𝑥
− 2
= 3 + 2 2 +
1
3 + 2 2
− 2

If x = 3 + 2 2, then 𝑥 −
1
𝑥
is
1
2
2 2
3 3
X = 3 + 2 2
𝑥 −
1
𝑥
2
= 𝑥 +
1
𝑥
− 2
= 3 + 2 2 +
1
3 + 2 2
− 2
= 3 + 2 2 +
1
3 + 2 2
𝑥
3 − 2 2
3 − 2 2

If x = 3 + 2 2, then 𝑥 −
1
𝑥
is
1
2
2 2
3 3
X = 3 + 2 2
𝑥 −
1
𝑥
2
= 𝑥 +
1
𝑥
− 2
= 3 + 2 2 +
1
3 + 2 2
− 2
= 3 + 2 2 +
1
3 + 2 2
𝑥
3 − 2 2
3 − 2 2
= 3 + 2 2 + (3 - 2 2) − 2

If x = 3 + 2 2, then 𝑥 −
1
𝑥
is
1
2
2 2
3 3
X = 3 + 2 2
𝑥 −
1
𝑥
2
= 𝑥 +
1
𝑥
− 2
= 3 + 2 2 +
1
3 + 2 2
− 2
= 3 + 2 2 +
1
3 + 2 2
𝑥
3 − 2 2
3 − 2 2
= 3 + 2 2 + (3 - 2 2) − 2
= 4

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