Surds and Indices Practice Test Questions – Aptitude -- Free Practice

Question 1

Solve for x: 2^x+3 − 2^x−1 = 120.

A

4✓✗

B

3✓✗

C

5✓✗

D

6✓✗

Explanation

Take 2^x−1 common: 2^x−1 (2⁴ − 1) = 120.
2^x−1 × 15 = 120, so 2^x−1 = 8 = 2³.
x − 1 = 3, so x = 4.

Shortcut Method

2^x+3 = 16 × 2^x−1
(16 − 1) × 2^x−1 = 120 → 2^x−1 = 8 = 2³
∴ x = 4 ✓

Question 2

Find the value of x if 3^x+1 − 3^x−1 = 72.

A

2✓✗

B

3✓✗

C

4✓✗

D

5✓✗

Explanation

Take 3^x−1 common: 3^x−1 (3² − 1) = 72.
3^x−1 × 8 = 72, so 3^x−1 = 9 = 3².
x − 1 = 2, so x = 3.

Shortcut Method

3^x+1 = 9 × 3^x−1
(9 − 1) × 3^x−1 = 72 → 3^x−1 = 9 = 3²
∴ x = 3 ✓

Question 3

What will be value of (m+n) if we know that

A

4✓✗

B

2✓✗

C

0✓✗

D

1✓✗

Explanation

Try 28 - 6√3 = (a - b√3)² = a² + 3b² - 2ab√3. So ab = 3 and a² + 3b² = 28, which gives a = 1 and b = 3.
So the root = |1 - 3√3| = 3√3 - 1. Comparing with √3 m + n gives m = 3 and n = -1.
m + n = 2.

Question 4

Find the value of M + 1/M (surds).

A

2✓✗

B

9✓✗

C

8✓✗

D

√15✓✗

Explanation

M = (√5 - √3) ÷ (√5 + √3), so 1M = (√5 + √3) ÷ (√5 - √3).
Over the common denominator (√5 + √3)(√5 - √3) = 5 - 3 = 2, the numerator = (√5 - √3)² + (√5 + √3)² = (8 - 2√15) + (8 + 2√15) = 16.
M + 1M = 162 = 8.

Question 5

What is the value of (0.000001)^1/3?

A

0.000001✓✗

B

0.001✓✗

C

1100✓✗

D

0.1✓✗

Explanation

0.000001 = (0.01)³, so (0.000001)^1/3 = 0.01 = 1100.

Question 6

If the value of √28 is approximately 5.2915, then the value of √74 is approximately

A

1.2000✓✗

B

0.5687✓✗

C

1.3229✓✗

D

1.4652✓✗

Explanation

√28 = 2√7, so √7 = 5.29152 = 2.64575.
√74 = √72 = 2.645752 = 1.3229 (approximately).

Question 7

How many zeroes are there in 2000¹⁰?

A

10✓✗

B

30✓✗

C

20✓✗

D

15✓✗

Explanation

2000¹⁰ = (2 × 10³)¹⁰ = 2¹⁰ × 10³⁰.
2¹⁰ = 1024 has no trailing zeros, so there are 30 zeros.

Question 8

What is value of M in (p/q)^2M+2 = (q/p)^9-M

A

6✓✗

B

5✓✗

C

-72✓✗

D

-11✓✗

Explanation

qp = (pq)^-1, so the right side = (pq)^{-(9 - M)}.
Equating the powers: 2M + 2 = M - 9, so M = -11.

Question 9

Find the value of X.
(15)^3.5 * (15)^x = 15⁸

A

2.29✓✗

B

2.75✓✗

C

4.25✓✗

D

4.5✓✗

Explanation

15^3.5 × 15^x = 15^{3.5 + x} = 15⁸, so 3.5 + x = 8.
x = 4.5.

Question 10

If 5^a = 625, then the value of 5^{(a – 2)} is:

A

5✓✗

B

25✓✗

C

125✓✗

D

625✓✗

Explanation

5ᵃ = 625 = 5⁴, so a = 4.
5^{a - 2} = 5² = 25.

Question 11

[ 2 + 2*√6 ] / 2 =

A

√6✓✗

B

2*√6✓✗

C

1 + √6✓✗

D

1 + 2*√6✓✗

Explanation

(2 + 2√6) ÷ 2 = 2(1 + √6) ÷ 2 = 1 + √6.

Question 12

If m is an integer such that (-2) ^2m = 2 ^9-m then m =

A

1✓✗

B

2✓✗

C

3✓✗

D

4✓✗

Explanation

(-2)^2m = 2^2m, because the power is even.
2^2m = 2^{9 - m}, so 2m = 9 - m and 3m = 9, so m = 3.

Question 13

110²² - 110²³ =

A

110✓✗

B

910²³✓✗

C

-110✓✗

D

-110²²✓✗

Explanation

110²² - 110²³ = 10 - 110²³ = 910²³.

Question 14

Find the value of 5^1/4 × 125^1/4.

A

5✓✗

B

25✓✗

C

50✓✗

D

10✓✗

Explanation

5^1/4 × 125^1/4 = (5 × 125)^1/4 = 625^1/4 = 5.

Question 15

(√3 + 2)(√3 - 2) = ?

A

√3 - 4✓✗

B

√6 - 4✓✗

C

-1✓✗

D

1✓✗

Explanation

Using the difference of squares: (√3)²-2² = 3-4 = -1.

Question 16

3^2n-1=127^n-3. Find n.

A

2✓✗

B

3✓✗

C

6✓✗

D

4✓✗

Explanation

127^n-3 = 3^-3(n-3) = 3^9-3n. So 2n-1=9-3n, giving 5n=10, n=2.

Question 17

(8125)^-4/3=?

A

1215✓✗

B

6425✓✗

C

62516✓✗

D

12581✓✗

Explanation

8125=(25)³. [(25)³]^-4/3=(25)^-4=(52)⁴=625/16.

Question 18

If xyz=1, find 1/(1+x+y^-1)+1/(1+y+z^-1)+1/(1+z+x^-1)

A

1✓✗

B

0✓✗

C

2✓✗

D

None of These✓✗

Explanation

This is a standard cyclic identity that equals 1 whenever xyz=1 (verified with x=2, y=12, z=1: terms are 15 + 25 + 25 = 1).

Question 19

2^x=cube root of 64. Find x.

A

3✓✗

B

23✓✗

C

2✓✗

D

4✓✗

Explanation

Cube root of 64 is 4. 2^x=4=2², so x=2.

Question 20

x^a=y^b=z^c and y²=xz. Find b.

A

2ac/(a+c)✓✗

B

x(a+c)/yz✓✗

C

(a+c)/ac✓✗

D

acxyz✓✗

Explanation

Let x^a=y^b=z^c=k. Then y²=xz gives k^2/b=k^1/a+1/c, so 2/b=(a+c)/(ac), giving b=2ac/(a+c).

Question 21

25^x+3=25^3x-4. Find x.

A

144✓✗

B

403✓✗

C

253✓✗

D

278✓✗

Explanation

Same base, so exponents are equal: x+3=3x-4, giving 7=2x, x=72=14/4.

Question 22

1/(1+x^a-b)+1/(1+x^b-a)=?

A

1✓✗

B

0✓✗

C

2✓✗

D

None of These✓✗

Explanation

Let y=x^a-b, so x^b-a=1/y. Expression = 1/(1+y) + y/(y+1) = (1+y)/(1+y) = 1.

Question 23

√2ⁿ=128. Find n.

A

24✓✗

B

14✓✗

C

36✓✗

D

15✓✗

Explanation

2^n/2=128=2⁷, so n/2=7, n=14.

Question 24

The value of (27216)^-2/3 is:

A

5✓✗

B

6✓✗

C

7✓✗

D

4✓✗

Explanation

27216 = (36)³ = (12)³. [(12)³]^-2/3 = (12)^-2 = 4.

Question 25

The value of (729)^1/3 is

A

11✓✗

B

9✓✗

C

10✓✗

D

3✓✗

Explanation

9³ = 729.