Index Notation

Dhristi JEE 2022-24

Index Notation Definition

Index notation is a method of representing numbers and letters that have been multiplied by themself multiple times.


For example, the number 360 can be written as either \[2 \times 2 \times 2 \times 3 \times 3 \times 5\] or \[2^{3} \times 3^{3} \times 5\]. 


\[2^{3}\] is read as ‘’2 to the power of 3” or “2 cubed” and means \[2 \times 2 \times 2 \].


\[3^{2}\] is read as ‘’3 to the power of 2” or “3 squared” and means \[3 \times 3 \].


\[ Y^{n}  = \frac{y*y*y*y*...*y*y}{"n"  "lots of"  "y"}\]


In general, yⁿ is read as “y to the power of n” and means "n lots of y", multiplied together”.


Numbers represented in index notation are often known as exponents or powers. In the above notation, y is the base number, and n is the exponent.


What is Known as an Index Number? 

An index number is defined as the number which is raised to the power. The power says how many times the number is to be used in multiplication.


Generally, it is represented as a small number to the right side and above the base number.


In the above example, the little “3” says to use 8 three times in multiplication. It is read as “ 8 to the power of 3”.


Index Notation Rules

Following are some of the exponent or index rules. These are basic rules of:

Rule 1: When two numbers with the same base are multiplied, their powers get added.

Example:

\[ 2^{4} * 2^{2} =(2*2*2*2)(2*2)\]

\[= (2*2*2*2*2*2)\]

\[ = 26 = 2^{(4+2)}\]


Rule 2: When two numbers with the same base are divided, their powers get subtracted.

Example:

\[ \frac{3^{5}}{3^{3}}= \frac{3*3*3*3*3}{3*3*3} = 3^{(5-3)} = 3^{2} = 9 \]


Rule 3: Any number raised to 0 is equal to 1.

Example:

70=1 or 80=1


Rule 4: If any term with power is raised to the exponent or power, the exponents or powers are multiplied together.

Example:

\[ (2^{2}) ^{2} = 2^{2*3} = 26\]


Rule 5: Any negative powers can be represented in a fractional form.

Example:

\[a - x = \frac{1}{ax}\]


Rule 6: The exponent or index given in a fraction form can be represented as the radical form.

Example:

\[ Y\frac{2}{3} = (\sqrt[3]{Y})^{2}\]


Power of 10

Power of 10 is a unique way of writing large numbers or smaller numbers. Instead of using so many zeroes, you can show how many powers of the 10 will make that many zeroes. For example, 6000 in the power of 10 can be written as:


\[6000=6*1000= 6* 10^{3}\]


6 thousand is 6 times a thousand. And, a thousand in 6000 is \[ 10^{3}\]. Hence 6 times \[ 10^{3} = 6000\].


Power of 10 is extremely used by Scientists and Engineers as they deal with the numbers that include large numbers of zeroes. For example, the mass of the Sun  that is 1988,000,000,000,000,000,000,000,000 kgs can be written in power of 10 as 1.988×1030


Index Notation Examples

Following are some of the index notation examples:


1. Express the prime factors of 98 in index notation form.

Solution:

Prime factors of 98 are = 2×7×7×7×7×7×7×7

Prime factors of 98 in index notation can be represented as 2×72


2. Evaluate \[ \frac{81}{16}^{-\frac{3}{4}}\]

Solution: 

\[ \frac{81}{16}^{-\frac{3}{4}}\]

\[ = \frac{1}{\frac{81}{16}^{\frac{3}{4}}}\]

\[ = (\frac{16}{81})^{\frac{3}{4}}\]

\[ = \frac{16}{81}(\frac{1}{4})^{3}\]

\[ = \frac{2}{3}^{3}\]

\[ = \frac{8}{27}\]


3. Evaluate \[2^{3}*3^{2}*5^{2}*3^{3}\]. Write the answer in index notational form.

Solution:

\[2^{3}*3^{2}*5^{2}*3^{3}\]

\[ = 2^{3}*3^{2+3}*5^{2}\]

\[ = 2^{3}*3^{5}*5^{2}\]


4. Determine 25÷23 and express the answers in index notation.

Solution: As we know, when two numbers with the same base are divided, their powers get subtracted.

Accordingly,

25÷23

=  22

=  4

The answer in index notation can be represented as 22. 


Did You Know? 

The distance light travels in one year can be easily calculated in the form of index notation as 9.461 × 10¹⁵.


Index Notation is also known as exponential form or exponential notation.


FAQs on Index Notation

1. What are Negative Exponents?

Negative exponent such as x-n = 1xn says that negative exponents in the numerator get moved to the denominator and become positive exponents whereas negative exponents in the denominator get moved to the numerator and become positive. Only the negative exponents are moved. For example,


6-2 = 162 = 136a-3b-7 = b7a3

2. Mention the Difference Between Exponents and Power.

In Mathematics, exponents means the small number, placed at the up-right of the whole number. It represents the number of times the base number is used as a factor in multiplying itself. For example, in 3³ = 3 × 3 × 3, 3 is the exponent which represents the number 3 is to be multiplied by itself.


Power, in Mathematics, refers to the whole number that denotes the repeated multiplication of the same number. For example, in 3³ = 3 × 3 × 3, 3 is the base number that is multiplying itself thrice and read as “ three to the power of three” or “ three to the third power.

3. Define Index.

Index of a variable or (constant) is the value that is raised to the power of variables. The indices are also known as powers or exponents. It states the number of times the given number has to be multiplied. It is represented in the form: 3³ = 3 × 3 × 3.

4. What is index notation? What are the laws of indices?

The method through which one can represent numbers as well as letters that have been multiplied by themselves more than once, is referred to as the method of index notation. It is very useful as it helps to specify the elements of various numbers. 

An algebraic expression is one wherein a particular quantity is made up of symbols together with other operations. And thus, the laws of indices are used to further simplify expressions that involve indices. These laws of indices are as follows: 

  • The first law is that of multiplication which states that two terms with the same base are supposed to be multiplied together, then their indices are added. 

  • The second law is that of division. It states that if two terms have the same base and are supposed to be divided together, then their indices are to be subtracted. This law is also helpful in explaining how anything to the power of zero, always equals one. 

  • The third law talks about brackets; if a term with a certain power is further, itself raised to a power, then these powers are multiplied together. 

  • There is yet another law that states that both the numerator as well as the denominator of a certain fractional power tend to hold some sort of meaning. 

5. Are exponents and powers the same thing?

No, exponents and powers are different concepts. While an exponent is a quantity that tends to represent the specific power to which a number is raised, power doesn’t function in the same way. Power is more of an expression that helps to represent the multiplication of the same number repeatedly. However, even though both their concepts are quite different from each other, both these terms (exponents and powers) are still used in mathematical operations interchangeably quite often.​​ 


The small number that is placed at the top right corner of the base number is an exponent, and while raising the base number to the exponent, the number that we get as a result is referred to as its power.

6. What do you mean by fractional indices? 

The given powers of some particular terms are known as fractional indices wherein both parts of the fraction, i.e., the numerator as well as the denominator, have meanings. For instance, let’s consider \[ x^{\frac{a}{b}}\]. Here, a, i.e., the numerator of the fraction, is considered to be the power to which the answer is to be raised. And b, i.e., the denominator of the fraction, is the root of the given letter or number. An important point to be noted here is that there is a difference between integer and fractional powers and that their concepts are not the same.

7. What happens if there are arrays with more than one index?

Besides arrays with just a single index, there are certainly other types too which are commonly known as multidimensional arrays. Let us take a two-dimensional table as an example. Now, there would be three possibilities regarding the same. They are as follows: 

  • By computing a single index out of the two, you can change the two-dimensional array into a one-dimensional one. 

  • There is a possibility of an array of arrays. In simpler terms, consider a one-dimensional array wherein each and every single element happens to be just another one-dimensional array. 

  • If you want to hold the addresses of each of the rows of the original array, you can make use of the additional storage to do that. As a result, you can then store the rows of these additional arrays in the form of separate one-dimensional arrays.

8. What is the difference between positive and negative exponents? Can you change a negative exponent to a positive one?

A negative exponent tells us just how many times the base number is to be divided. A positive exponent, on the other hand, shows us how many times that base number is to be multiplied. And according to the negative exponent rule, a number that has a negative exponent is supposed to be put to the denominator and vice versa.

  

For changing a negative exponent into a positive one, the reciprocal of the given number is to be written.

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