Surds And Indices

Surds

A surd is a square root which cannot be reduced to a rational number.

For example, is not a surd.

However is a surd.

If you use a calculator, you will see that and we will need to round the answer correct to a few decimal places. This makes it less accurate.

If it is left as , then the answer has not been rounded, which keeps it exact.

Here are some general rules when simplifying expressions involving surds.

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a^m x aⁿ = a^m⁺ⁿ

a^m	= a^m^–ⁿ
aⁿ	= a^m^–ⁿ

(a^m)ⁿ = a^mn

(ab)ⁿ = aⁿbⁿ

a	n	=	aⁿ
b	n	=	bⁿ

a⁰ = 1

Questions

Level-I

(17)^3.5 x (17)^? = 17⁸

A.	2.29
B.	2.75
C.	4.25
D.	4.5

If		a		x – 1	=		b		x – 3	, then the value of x is:
		b					a

Given that 10^0.48 = x, 10^0.70 = y and x^z = y², then the value of z is close to:

A.	1.45
B.	1.88
C.	2.9
D.	3.7

If 5^a = 3125, then the value of 5⁽^a^{– 3)} is:

A.	25
B.	125
C.	625
D.	1625

If 3⁽^x^–^y⁾ = 27 and 3⁽^x⁺^y⁾ = 243, then x is equal to:

A.	0
B.	2
C.	4
D.	6

.6.

(256)^0.16 x (256)^0.09 = ?

A.	4
B.	16
C.	64
D.	256.25

The value of [(10)¹⁵⁰ ÷ (10)¹⁴⁶]

A.	1000
B.	10000
C.	100000
D.	10⁶

1	+	1	+	1	= ?
1 + x⁽^b^–^a⁾ + x⁽^c^–^a⁾		1 + x⁽^a^–^b⁾ + x⁽^c^–^b⁾		1 + x⁽^b^–^c⁾ + x⁽^a^–^c⁾

A.	0
B.	1
C.	x^a^–^b^–^c
D.	None of these

(25)^7.5 x (5)^2.5 ÷ (125)^1.5 = 5^?

A.	8.5
B.	13
C.	16
D.	17.5
E.	None of these

10.

(0.04)^-1.5 = ?

A.	25
B.	125
C.	250
D.	625

Level-II

11.

(243)ⁿ^/5 x

3²ⁿ^{+ 1}

= ?

9ⁿ x 3ⁿ^{– 1}

A.	1
B.	2
C.	9
D.	3ⁿ

12.

1	+	1	= ?
1 + a⁽ⁿ^–^m⁾		1 + a⁽^m^–ⁿ⁾

a^m⁺ⁿ

13.

If m and n are whole numbers such that mⁿ = 121, the value of (m – 1)ⁿ^{+ 1} is:

A.	1
B.	10
C.	121
D.	1000

14.

x^b

(b + c – a)

x^c

(c + a – b)

x^a

⁽^a⁺^b^–^c⁾

= ?

x^c

x^a

x^b

A.	x^abc
B.	1
C.	x^ab⁺^bc⁺^ca
D.	x^a⁺^b⁺^c

15. If 5√5 * 5³ ÷ 5^-3/2= 5^a+2 , the value of a is:
A. 4
B. 5
C. 6
D. 8

16.(132)⁷ ×(132)^? =(132)^11.5.

A. 3
B. 3.5
C. 4
D. 4.5

17. (ab)x−2=(ba)x−7. What is the value of x ?

A. 3
B. 4
C. 3.5
D. 4.5

18. (0.04)^-2.5 = ?

A. 125
B. 25
C. 3125
D. 625

Answers

Level-I

Answer:1 Option D

Explanation:

Let (17)^3.5 x (17)^x = 17⁸.

Then, (17)^{3.5 +}^x = 17⁸.

3.5 + x = 8

x = (8 – 3.5)

x = 4.5

Answer:2 Option C

Explanation:

Given	a		x – 1	=		b		x – 3
	b					a

x – 1

-(x – 3)

(3 – x)

x – 1 = 3 – x

2x = 4

x = 2.

Answer:3 Option C

Explanation:

x^z = y² 10^(0.48^z⁾ = 10^{(2 x 0.70)} = 10^1.40

0.48z = 1.40

z =	140	=	35	= 2.9 (approx.)
	48		12

Answer:4 Option A

Explanation:

5^a = 3125 5^a = 5⁵

a = 5.

5⁽^a^{– 3)} = 5^{(5 – 3)} = 5² = 25.

Answer:5 Option C

Explanation:

3^x^–^y = 27 = 3³ x – y = 3 ….(i)

3^x⁺^y = 243 = 3⁵ x + y = 5 ….(ii)

On solving (i) and (ii), we get x = 4

Answer:6 Option A

Explanation:

(256)^0.16 x (256)^0.09 = (256)^{(0.16 + 0.09)}

= (256)^0.25

= (256)^(25/100)

= (256)^(1/4)

= (4⁴)^(1/4)

= 4^4(1/4)

= 4¹

= 4

Answer:7 Option B

Explanation:

(10)¹⁵⁰ ÷ (10)¹⁴⁶ =	10¹⁵⁰
	10¹⁴⁶

= 10^{150 – 146}

= 10⁴

= 10000.

Answer:8 Option B

Explanation:

Given Exp. =

	1 +	x^b	+	x^c
		x^a		x^a

	1 +	x^a	+	x^c
		x^b		x^b

	1 +	x^b	+	x^a
		x^c		x^c

=	x^a	+	x^b	+	x^c
	(x^a + x^b + x^c)		(x^a + x^b + x^c)		(x^a + x^b + x^c)

=	(x^a + x^b + x^c)
	(x^a + x^b + x^c)

= 1.

Answer:9 Option B

Explanation:

Let (25)^7.5 x (5)^2.5 ÷ (125)^1.5 = 5^x.

Then,	(5²)^7.5 x (5)^2.5	= 5^x
	(5³)^1.5

	5^{(2 x 7.5)} x 5^2.5	= 5x
	5^{(3 x 1.5)}

	5¹⁵ x 5^2.5	= 5^x
	5^4.5

5^x = 5^{(15 + 2.5 – 4.5)}

5^x = 5¹³

x = 13.

Answer:10 Option B

Explanation:

(0.04)^-1.5 =		4		-1.5
		100

=		1		-(3/2)
		25

= (25)^(3/2)

= (5²)^(3/2)

= (5)^{2 x (3/2)}

= 5³

= 125.

Level-II

Answer:11 Option C

Explanation:

Given Expression

=	(243)⁽ⁿ^/5) x 3²ⁿ^{+ 1}
	9ⁿ x 3ⁿ^{– 1}

=	(3⁵)⁽ⁿ^/5) x 3²ⁿ^{+ 1}
	(3²)ⁿ x 3ⁿ^{– 1}

=	(3^{5 x (}ⁿ^/5) x 3²ⁿ^{+ 1})
	(3²ⁿ x 3ⁿ^{– 1})

=	3ⁿ x 3²ⁿ^{+ 1}
	3²ⁿ x 3ⁿ^{– 1}

=	3⁽ⁿ^{+ 2}ⁿ^{+ 1)}
	3⁽²ⁿ⁺ⁿ^{– 1})

=	3³ⁿ^{+ 1}
	3³ⁿ^{– 1}

= 3⁽³ⁿ^{+ 1 – 3}ⁿ^{+ 1)} = 3² = 9.

Answer:12 Option C

Explanation:

	1 +	aⁿ
		a^m

	1 +	a^m
		aⁿ

1 + a⁽ⁿ^–^m⁾

1 + a⁽^m^–ⁿ⁾

=	a^m	+	aⁿ
	(a^m + aⁿ)		(a^m + aⁿ)

=	(a^m + aⁿ)
	(a^m + aⁿ)

= 1.

Answer:13 Option D

Explanation:

We know that 11² = 121.

Putting m = 11 and n = 2, we get:

(m – 1)ⁿ^{+ 1} = (11 – 1)^{(2 + 1)} = 10³ = 1000.

Answer:14 Option B

Explanation:

Given Exp.

= x⁽^b^–^c⁾⁽^b⁺^c^–^a⁾ . x⁽^c^–^a⁾⁽^c⁺^a^–^b⁾ . x⁽^a^–^b⁾⁽^a⁺^b^–^c⁾

= x⁽^b^–^c⁾⁽^b⁺^c^{) –}^a⁽^b^–^c⁾ . x⁽^c^–^a⁾⁽^c⁺^a^{) –}^b⁽^c^–^a⁾
. x⁽^a^–^b⁾⁽^a⁺^b^{) –}^c⁽^a^–^b⁾

= x⁽^b^2 –^c^2 +^c^2 –^a^2 +^a^2 –^b²⁾ . x^–^a⁽^b^–^c^{) –}^b⁽^c^–^a^{) –}^c⁽^a^–^b⁾

= (x⁰ x x⁰)

= (1 x 1) = 1.

Answer:15 option C

Answer:16

Explanation

am.an=am+n

(132)⁷ × (132)^x = (132)^11.5

=> 7 + x = 11.5

=> x = 11.5 – 7 = 4.5

Answer:17

Explanation:

an=1a−n

(ab)x−2=(ba)x−7⇒(ab)x−2=(ab)−(x−7)⇒x−2=−(x−7)⇒x−2=−x+7⇒x−2=−x+7⇒2x=9⇒x=92=4.5

Answer:18

Explanation:

a−n=1/an

(0.04)−2.5=(1/.04)2.5=(100/4)2.5=(25)2.5=(52)2.5=(52)(5/2)=55=3125