SURDS

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.

 

 

 

  1. aman = am + n
amam – n
an
  
  • (am)namn

 

  1. (ab)nanbn

 

an=an
bbn
      
  1. a0= 1

 

 

Questions

Level-I

 

1.(17)3.5 x (17)? = 178
A.2.29
B.2.75
C.4.25
D.4.5

 

2.
Ifax – 1=bx – 3, then the value of x is:
ba
A.
1
2
B.1
C.2
D.
7
2

 

3.Given that 100.48 = x, 100.70 = y and xz = y2, then the value of z is close to:
A.1.45
B.1.88
C.2.9
D.3.7

 

4.If 5a = 3125, then the value of 5(a – 3) is:
A.25
B.125
C.625
D.1625

 

5.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

 

7.The value of [(10)150 ÷ (10)146]
A.1000
B.10000
C.100000
D.106

 

8.
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.xa – b – c
D.None of these

 

9.(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.0
B.
1
2
C.1
D.am + 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.
xb(b + c – a).xc(c + a – b).xa(a + b – c)= ?
xcxaxb
A.xabc
B.1
C.xab + bc + ca
D.xa + b + c

 

  1. If 5√5 * 53÷ 5-3/2= 5a+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:

Givenax – 1=bx – 3
ba

 

ax – 1=a-(x – 3) =a(3 – x)
bbb

x – 1 = 3 – x

2x = 4

x = 2.

 

 

Answer:3 Option C

 

Explanation:

xz = y2        10(0.48z) = 10(2 x 0.70) = 101.40

0.48z = 1.40

 z =140=35= 2.9 (approx.)
4812

 

Answer:4 Option A

 

Explanation:

5a = 3125        5a = 55

a = 5.

5(a – 3) = 5(5 – 3) = 52 = 25.

 

 

Answer:5 Option C

 

Explanation:

3x – y = 27 = 33        x – y = 3 ….(i)

3x 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)

= 44(1/4)

= 41

= 4

Answer:7 Option B

 

Explanation:

(10)150 ÷ (10)146 =10150
10146

= 10150 – 146

= 104

= 10000.

 

Answer:8 Option B

 

Explanation:

Given Exp. =
1 +1 +1
1 +xb+xc
xaxa
1 +xa+xc
xbxb
1 +xb+xa
xcxc

 

   =xa+xb+xc
(xa + xb + xc)(xa + xb + xc)(xa + xb + xc)

 

   =(xa + xb + xc)
(xa + xb + xc)

= 1.

 

Answer:9 Option B

 

Explanation:

Let (25)7.5 x (5)2.5 ÷ (125)1.5 = 5x.

Then,(52)7.5 x (5)2.5= 5x
(53)1.5

 

5(2 x 7.5) x 52.5= 5x
5(3 x 1.5)

 

515 x 52.5= 5x
54.5

5x = 5(15 + 2.5 – 4.5)

5x = 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
=(35 x (n/5) x 32n + 1)
(32n x 3n – 1)
=3n x 32n + 1
32n x 3n – 1
=3(n + 2n + 1)
3(2n + n – 1)
=33n + 1
33n – 1
= 3(3n + 1 – 3n + 1)   = 32   = 9.

Answer:12 Option C

 

Explanation:

1+1=
1 +1
1 +an
am
1 +am
an
1 + a(n – m)1 + a(m – n)

 

   =am+an
(am + an)(am + an)

 

   =(am + an)
(am + 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(b2 – c2 + c2 – a2 + a2 – b2)  .   xa(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

 ,

SURDS is an acronym for the following subtopics of computer science:

  • Abstract data types (ADTs) are a way of defining data structures and their operations in a way that is independent of their implementation. This allows programmers to focus on the logical properties of data structures, without worrying about how they are actually implemented.
  • Algorithms are step-by-step procedures for solving problems. They are often used to find the most efficient way to solve a problem.
  • Complexity theory is the study of how the time and space requirements of algorithms grow as the input size increases.
  • Data structures are ways of organizing data so that it can be stored and accessed efficiently.
  • Databases are collections of data that are stored and organized in a way that makes it easy to retrieve and update the data.
  • Discrete mathematics is the study of mathematical structures that are discrete, as opposed to continuous. This includes topics such as sets, relations, functions, graphs, and logic.
  • Information retrieval is the process of finding information from a collection of documents. This can be done using a variety of methods, such as keyword search, full-text search, and information filtering.
  • Machine Learning is a field of computer science that gives computers the ability to learn without being explicitly programmed. This is done by using algorithms that can learn from data.
  • Natural language processing is a field of computer science that deals with the interaction between computers and human language. This includes tasks such as text analysis, machine translation, and speech recognition.
  • Operating systems are Software that control the hardware and Resources of a computer. They provide a platform for other software to run on, and they manage the computer’s resources such as memory, CPU, and storage.
  • Programming languages are a set of instructions that allow programmers to write code that can be executed by a computer. There are many different programming languages, each with its own strengths and weaknesses.
  • Software engineering is the discipline of designing, developing, and maintaining software. It is a complex field that involves a variety of skills, such as problem solving, algorithm design, data structures, and software architecture.
  • Systems programming is a type of programming that focuses on the design and implementation of operating systems, device drivers, and other low-level software. It is a challenging field that requires a deep understanding of computer architecture and operating systems.
  • Theoretical computer science is the study of the theoretical foundations of computer science. It includes topics such as computability theory, automata theory, and cryptography.

These are just a few of the many subtopics of computer science. It is a vast and ever-growing field, with new developments happening all the time. If you are interested in learning more about computer science, there are many resources available online and in libraries.

What is a prime number?

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number.

What is a composite number?

A composite number is a natural number greater than 1 that is not prime. A natural number greater than 1 that is not prime is called a composite number.

What is a factor of a number?

A factor of a number is a natural number that divides evenly into that number. For example, 1, 2, 3, 4, 6, and 12 are all factors of 12.

What is a multiple of a number?

A multiple of a number is a number that can be obtained by multiplying that number by a natural number. For example, 2, 4, 6, 8, 10, and 12 are all multiples of 6.

What is a prime factorization?

A prime factorization of a number is a way of writing that number as a product of prime numbers. For example, the prime factorization of 12 is 2 x 2 x 3.

What is a greatest common factor?

The greatest common factor (GCF) of two or more numbers is the largest number that is a factor of all of them. For example, the GCF of 12 and 18 is 6.

What is a least common multiple?

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of them. For example, the LCM of 12 and 18 is 36.

What is a decimal?

A decimal is a number that can be written in the form $a.b$, where $a$ is an integer and $b$ is a non-negative integer less than 1. For example, 0.5, 1.2, and 3.14 are all decimals.

What is a fraction?

A fraction is a part of a whole. It is written as two numbers, one on top of the other, with a line between them. The number on the top is called the numerator, and the number on the bottom is called the denominator. For example, $\frac{1}{2}$ is a fraction that represents one part of a whole that has been divided into two parts.

What is a Percentage?

A percentage is a number or ratio that is expressed as a fraction of 100. It is written with a percent sign (%). For example, 50% is equivalent to $\frac{50}{100}$, which is also equal to 0.5.

What is a ratio?

A ratio is a comparison of two quantities. It is written as two numbers, one on top of the other, with a colon between them. For example, the ratio of 2 to 3 can be written as $2:3$.

What is a proportion?

A proportion is an equation that states that two ratios are equal. For example, the proportion $\frac{2}{3} = \frac{4}{6}$ states that the ratio of 2 to 3 is equal to the ratio of 4 to 6.

What is a linear equation?

A linear equation is an equation in which the highest power of the variable is 1. For example, the equation $y = 2x + 1$ is a linear equation.

What is a quadratic equation?

A quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation $y = x^2 + 2x – 3$ is a quadratic equation.

What is a cubic equation?

A cubic equation is an equation in which the highest power of the variable is 3. For example, the equation $y = x^3 – 2x^2 + 3x – 1$ is a cubic equation.

What is a radical equation?

A radical equation is an equation that contains one or more radicals. For example, the equation $x = \sqrt{2}$ is a radical equation.

What is an exponential equation?

An exponential equation is an equation in which one variable is raised to the power of another variable. For example, the equation $y = 2^x$ is an exponential equation.

What is a logarithmic equation?

A logarithmic equation is an equation in which one

Sure. Here are some MCQs without mentioning the topic SURDS:

  1. Which of the following is not a rational number?
    (A) $\frac{1}{2}$
    (B) $\frac{3}{4}$
    (C) $\pi$
    (D) $\sqrt{2}$

  2. Which of the following is not an integer?
    (A) 1
    (B) 2
    (C) $\pi$
    (D) -5

  3. Which of the following is not a whole number?
    (A) 0
    (B) 1
    (C) $\frac{1}{2}$
    (D) 2

  4. Which of the following is not a natural number?
    (A) 1
    (B) 2
    (C) 3
    (D) $\pi$

  5. Which of the following is not a prime number?
    (A) 2
    (B) 3
    (C) 5
    (D) 7

  6. Which of the following is not a composite number?
    (A) 4
    (B) 6
    (C) 8
    (D) 9

  7. Which of the following is not a perfect square?
    (A) 1
    (B) 4
    (C) 9
    (D) 16

  8. Which of the following is not a perfect cube?
    (A) 1
    (B) 8
    (C) 27
    (D) 64

  9. Which of the following is not a perfect fifth?
    (A) 1
    (B) 3
    (C) 5
    (D) 7

  10. Which of the following is not a perfect fourth?
    (A) 1
    (B) 2
    (C) 3
    (D) 4

I hope these questions are helpful!