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300+ Power, Indices And Surds Question With Solution Free PDF – Download Free
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- Zakariyah, PhD Most of the equations we encounter, either in simplified or complex forms, are based on them. In fact they are part of our daily activities, e. So here we will take a look at how to simplify exponential and logarithmic expressions and how to solve their equations using certain rules that will be discussed shortly. However, graphs of their functions will not be covered here as these will be appropriately dealt with in another book. Readers are reminded that this part of the book focuses on the background regarding the topic and therefore few examples will be provided. For comprehensive and varied examples, do refer to the Worked Examples part pp.
- What is Indices? Well, indices is when a number is expressed in the form where is called the base and the index. The index, i. Nonetheless you probably would have found out that one of the terms index, exponent or power is used more than the others. In general, power is the most frequently used as such, is read as ' raised to the power of ', ' raised to the power ' or simply ' to the power. What does this imply? It means that has been multiplied by itself times. For example, if and then one can write read as 'five to the power three' or 'five cubed'. This implies , which equals. So what do you think about? It is in value but it is written in index form as. Now let us work in the reverse order and find out what is. This can be written as a multiple of tens, i. From this example, it is obvious that we need indices to express numbers such as in the example not only because it saves space and time but more importantly it is sometimes easy to remember and work with.
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So is a better way of writing a cumbersome 1 Calculation involving indices can be carried out on a scientific calculator, commonly via a button marked or something similar. The next question that might come to mind is whether any number can be written in index form. The answer is yes. Any rational number can be expressed in this form just like any integer can be written in the form where. For instance, can be expressed in both index and fractional forms as and respectively, but this is generally omitted.- This is the same for any other number. Laws of IndicesUndoubtedly one would need to carry out calculations in index form because it is a useful and compact way to express numbers. There are certain laws 2 that govern these operations and they will be discussed in this section. This can alternatively be expressed as The following should be noted about this rule:i The terms must have the same base otherwise the law cannot be used. For example because 5 and 6 are not like base. In other words, but. This can easily be proven because and. Again, this can alternatively be expressed asThe following should be noted about this rule:i The terms must have the same base otherwise the law cannot be used. For example. Again as previously performed and and. Law 3 Power LawThis law states that For example,.
- Alternatively, this can be shown using Law 1 as [ ] [ ] Special or derived lawsOther laws of indices include Law 4 Zero Power LawThis law can be written as It is simply put as 'anything' 3 to the power of zero is 1. Although this rule might look odd, it is logical and well established. This can be derived from Law 2. Sometimes one might need to carry out some operations that require a change from a positive index to a negative index. Let us account for this and re-state the rule as 'whenever the reciprocal of a number with an index is taken, the sign of the index changes'. For instance, or or In each of the above three cases, we have taken the reciprocal of the left-hand side to obtain the corresponding right-hand side. If it is not clear, this will shortly be put this into context. But 6 remember that a power is the inverse of a root just like a whole number is the opposite of a fraction.
- We know that the answers are and respectively. One can however obtain the same answers using Laws 2 and 3. Here we go: using Law 7 Same Power LawSo far we have introduced rules for dealing with terms that share a common base. We will now look at rules to be used when the bases of the terms are not the same. In general, the operations are carried out as you would normally do with one exception. This exclusion is that when the index of the terms is the same then the following rules should be applied: Before proceeding to logarithms, you may wish to go through the Worked Examples on indices first. These are contained in sections pp. What is Logarithm? Logarithm is a derived term from two Greek words, namely: logos expression and arithmos number Singh, Thus, logarithm is a technique of expressing numbers.
- In fact, it is a system of evaluating multiplication, division, powers and roots by appropriately converting them to addition and subtraction. This concept is primarily attributed to a Scotsman from Edinburgh, John Napier Bird, who developed it. Technically, the logarithm 6 of a number to a given base is the value of the power to which the base must be raised in order to produce the number. Let for and be the number 7 and its associated base respectively and equals to 'the logarithm to base of ' or 'the logarithm of to base ', then we can write this as 6 This is commonly shortened to or written as log and sometimes as lg.
- This is because raised to the power of anything is and so will only be valid for since. For example, the logarithm of to base 10 is 2 because, if the base 10 is raised to the power of 2 we will get the number In other words,. It is evident that indices and logarithm are related and one can change from one form to the other, i. This relationship can be written as Logarithms are classified according to the value of their base.
- Essentially, there are two or three types. In general, when the base is 10, it is usually omitted. In other words, is simply written as. Common logarithm is also called Briggsian named after Henry Briggs. Why 'natural'? It is possible that this logarithm is classified as natural due to the behaviour of certain natural phenomena e. Natural logarithm is also called hyperbolic or Napierian logarithm, named after its inventor John Napier. Furthermore, the logarithmic ratio unit, neper abbreviated as nep , is also used in his honour 8. Usually, natural logarithm is written as read as ' ' instead of as one might have expected. Laws of LogarithmLike indices, there are certain laws governing the operation of logarithms and these will be discussed under the following headings. Fundamental lawsEssentially, there are three main laws of logarithms. Law 1 Addition-Product LawThis rule can be written as In other words, the sum of logs of numbers to the same base is equal to the log of their products and vice versa.
- The following should be noted about the rule:i The logs must have the same base otherwise the law cannot be used. This can simply be proven as with the indices. Law 2 Subtraction-Quotient LawThis rule can be written as The following should be noted about the rule:i The logs must have the same base otherwise the law cannot be used. Again, this can be proven. Law 3 Power LawThis rule can be written asThere are three special formulae or properties resulting from the above power law, namely:, and Special or derived lawsOther laws of logarithms include Law 4 Unity Law or Log of Unity Law This rule can be written as This rule states that the logarithm of unity 1 to any base is zero.
Form 2 Questions And Answers On Indices And Logarithms - Tutorke
AllIndiaExams provides you lots of fully solved Aptitude Indices and Surds questions and answers with explanation. Fully solved examples with detailed answer description. Here you can find both objective type and Multiple Choice Aptitude Indices and Surds questions and answers for interview and entrance examination. True or false type questions are also provided here. Practice these bits well to get success in interviews or entrance tests How to solve these Aptitude Indices and Surds problems?- You no need to worry, we have given lots of solved Aptitude Indices and Surds questions with detailed explanation and also we have provided lots of short cut methods to quickly solve the questions within short time. So you can easily solve all kind of questions. Home Aptitude Indices and Surds Indices and Surds - Aptitude Questions and Answers This is aptitude questions and answers section on Indices and Surds with explanation for various interview, competitive examinations and entrance tests. Candidates who are attempting Indices and Surds Online test they can know their performance levels.
- These Indices and Surds Aptitude Multiple Choice Questions will prepare you for a technical job interview, as well as written test, and many certification exams.
- Thus, the missing number in the sequence is Hence D is the correct answer. Solution: C Explanation: The combination as a whole is transformed through rotation and reflection to find the best fit. However, it is apparent that for both choices A and B, the relative position of all parts are the same, but the inner white part-of-the-circle changes position. This is not the case with option C, where the relative position of all the constituent figure parts are the same as the entire set is rotated through degrees. Hence C is the correct answer. Sample Q 8 Identify the reflection of the given figure from the provided options. Solution: A Explanation: Since the original figure has arrows pointing in the north-east direction, the reflection should have arrows pointing towards the south-east direction.
- Now, in a reflection, the pentagons would be mirrored and there should not be any shift in relative position. Thus, out of A and B, B is eliminated. Option A satisfies all conditions of a reflection, mirrored direction but no shift in relative position otherwise. Hence the answer. Sample Q 9 Identify the next figure in the following sequence: Solution: C.
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Form 2 Questions And Answers On Indices And Logarithms - Tutorke
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