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.
- amx an = am + n
am | = am – n |
an | |
- (am)n= amn
- (ab)n= anbn
a | n | = | an | ||
b | bn | ||||
- a0= 1
Questions
Level-I
1. | (17)3.5 x (17)? = 178 | |||||||
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2. |
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3. | Given that 100.48 = x, 100.70 = y and xz = y2, then the value of z is close to: | |||||||
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4. | If 5a = 3125, then the value of 5(a – 3) is: | |||||||
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5. | If 3(x – y) = 27 and 3(x + y) = 243, then x is equal to: | |||||||
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.6. | (256)0.16 x (256)0.09 = ? | |||||||
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7. | The value of [(10)150 ÷ (10)146] | |||||||
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8. |
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9. | (25)7.5 x (5)2.5 ÷ (125)1.5 = 5? | |||||||||
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10. | (0.04)-1.5 = ? | |||||||
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Level-II
11. |
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12. |
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13. | If m and n are whole numbers such that mn = 121, the value of (m – 1)n + 1 is: | |||||||
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14. |
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- If 5√5 * 53÷ 5-3/2= 5a+2 , the value of a is:
A. 4
B. 5
C. 6
D. 8
A. 3
17. (ab)x−2=(ba)x−7. What is the value of x ?
A. 3
18. (0.04)-2.5 = ?
A. 125
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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 |
a | x – 1 | = | a | -(x – 3) | = | a | (3 – x) | |||||||
b | b | b |
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.) |
48 | 12 |
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. = |
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= | 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 |
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Answer:12 Option C
Explanation:
1 | + | 1 | = |
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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. |
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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
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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) -5Which of the following is not a whole number?
(A) 0
(B) 1
(C) $\frac{1}{2}$
(D) 2Which 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) 7Which of the following is not a composite number?
(A) 4
(B) 6
(C) 8
(D) 9Which of the following is not a perfect square?
(A) 1
(B) 4
(C) 9
(D) 16Which of the following is not a perfect cube?
(A) 1
(B) 8
(C) 27
(D) 64Which of the following is not a perfect fifth?
(A) 1
(B) 3
(C) 5
(D) 7Which of the following is not a perfect fourth?
(A) 1
(B) 2
(C) 3
(D) 4
I hope these questions are helpful!