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.

a^mx 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 3^{2n + 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

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^(0.48z) = 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 3^{2n + 1}
	9ⁿ x 3ⁿ^{– 1}

=	(3⁵)^(n/5) x 3^{2n + 1}
	(3²)ⁿ x 3ⁿ^{– 1}

=	(3^{5 x (n/5)} x 3^{2n + 1})
	(3²ⁿ x 3ⁿ^{– 1})

=	3ⁿ x 3^{2n + 1}
	3²ⁿ x 3ⁿ^{– 1}

=	3^{(n + 2n + 1)}
	3^{(2n + n – 1})

=	3^{3n + 1}
	3^{3n – 1}

= 3^{(3n + 1 – 3n + 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^{(b2 – c2 + c2 – a2 + a2 – 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

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:

Which of the following is not a rational number?
(A) $\frac{1}{2}$
(B) $\frac{3}{4}$
(C) $\pi$
(D) $\sqrt{2}$
Which of the following is not an integer?
(A) 1
(B) 2
(C) $\pi$
(D) -5
Which of the following is not a whole number?
(A) 0
(B) 1
(C) $\frac{1}{2}$
(D) 2
Which of the following is not a natural number?
(A) 1
(B) 2
(C) 3
(D) $\pi$
Which of the following is not a prime number?
(A) 2
(B) 3
(C) 5
(D) 7
Which of the following is not a composite number?
(A) 4
(B) 6
(C) 8
(D) 9
Which of the following is not a perfect square?
(A) 1
(B) 4
(C) 9
(D) 16
Which of the following is not a perfect cube?
(A) 1
(B) 8
(C) 27
(D) 64
Which of the following is not a perfect fifth?
(A) 1
(B) 3
(C) 5
(D) 7
Which of the following is not a perfect fourth?
(A) 1
(B) 2
(C) 3
(D) 4

I hope these questions are helpful!