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In Mathematics, Index (indices)Â is the power or exponent which is raised to a number or a variable. For example, in number 25, 5 is the index of 2. The plural form of index is indices. In algebra, we come across constants and variables. The constant value is fixed that cannot be changed, whereas a variable can be assigned any number or we can say its value can be changed. In algebra, mostly we deal with indices in terms of numbers.

Index of a variable is a value that is raised to the power of the variable. The index value is also known as powers or exponents. It shows the number of times the given number has to be multiplied. It is represented in the form:

a^{m} = a Ã— a Ã— a Ã— a .... Ã— a(m times)

Here, a is called base and m is the index.

The index says that a particular number (or base) is to be multiplied by itself or the number of times equal to the index raised to it. It is an easy way of writing big numbers and calculations.

There are some fundamental rules or laws of indices that are necessary to understand before we start solving problems with indices.Â We follow these rules while performing algebraic operations on indices. Following are the rules we need to follow:

Rule 1: If a constant or variable has index as â€˜0â€™, then the value of the result will be one. The base can be of any value.

a^{0} = 1

Example: 5^{0} = 1,12^{0} = 1

Rule 2: If the index value is negative, then it can be shown as the reciprocal of the positive index raised to the same variable.

a^{-p} = 1/a^{p}

Example: 5^{-1} = 1/5

Rule 3: If we multiply two variables with the same base, we need to add their powers and raise them to that base.

a^{p}.a^{q} = a^{p+q}

Example: 5^{2}.5^{3} = 5^{2+3} = 5^{5}

Rule 4: To divide two variables with the same base, we need to subtract the power of the denominator from the power of the numerator and raise it to that base.

a^{p}/a^{q} = a^{p-q}

Example:10^{4}/10^{3} = 10^{1}

Rule 5: When a variable with some index is again raised with a different index, then both the indices are multiplied together raised to the power of the same base.

(a^{p})^{q} = a^{p}^{q}

Example: (8^{2})^{4} = 8^{8}

Rule 6: When two variables with different bases, but the same indices are multiplied together, we have to multiply its base and raise the same index to multiplied variables.

a^{p}.b^{p} = (ab)^{p}

Example:3^{2}.4^{2} = (3.4)^{2} = 12^{2}

Rule 7: When two variables with different bases are given, and the same indices are divided.Â We have to divide the bases and raise the same index to it.

a^{p}/b^{p} = (a/b)^{p}

Rule 8:Â If an index is in the form of a fraction it can be represented in the radical form.

\[a^{\frac{p}{q}} = \sqrt[q]{a^{p}}\]

Example: \[5^{\frac{1}{2}} = \sqrt{5}\]

2 to the power of 10 = 210 = 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 = 1024.

"Powers of 10" is a very useful tool when we have to express large or small numbers.

Instead of having lots of zeros, we can show how many powers of 10 will make that many zeros.

Example: 5000 = 5 Ã— 1000 = 5 Ã— 10^{3}

5 thousand is 5 times a thousand. And a thousand is 10^{3}. So 5 times 10^{3} = 5,000

10 to the power of 3 = 10 Ã— 10 Ã— 10 = 1000

In words, we can say 10^{3} could be called "10 to the third power", "10 to the power 3" or simply "10 cubed".

10 to the power of 4 equals 10 Ã— 10 Ã— 10 Ã— 10 = 10000.

In words, we can say 10^{4} could be called "10 to the fourth power", "10 to the power 4" or "10 to the 4"

10 to the power of 6 = 10 Ã— 10 Ã— 10 Ã— 10 Ã— 10 Ã— 10 = 1000000

Negative Powers of 10

A negative power means how many times we have to divide by the number.

Example: 10 Ã· 10^{3} = 0.01

Other Way of Writing Ã— symbol

Sometimes people use the ^ symbol instead of x, as it is easy to type.

Example: 3 Ã— 10^4 is same as the 3 Ã— 10^{4}

3 Ã— 10^4 = 3 Ã— 10 Ã— 10 Ã— 10 Ã— 10 = 30,000

Note: For negative powers of 10: For negative powers of 10, move the decimal point to the left.

1. Find the value of \[27^{\frac{2}{3}}\]

Sol: Here index is in the form of a fraction. So we need to change it to radical form and then solve

\[27^{\frac{2}{3}} = \sqrt[3]{27^{2}}\]

27 can be written as 3^{3}

= \[\sqrt[3]{3^{6}}\]

= \[3^{\frac{6}{3}}\]

= 3^{2}

= 9

Hence the value of \[27^{\frac{2}{3}}\] is 9.

2. Express the value of 2^{5} Ã· 2^{2} in index notation.

Sol: According to the rule when two numbers with the same base are divided, we need to subtract the power of the denominator.

2^{5} Ã· 2^{2} = 2^{5-2} = 2^{3} = 8

Hence, the value of 2^{5} Ã· 2^{2} in index notation is 2^{3}.

The index of 10 means many places we have to move the decimal point. A positive value means to move it to the right, whereas negative means to the left. An example is given below:

FAQ (Frequently Asked Questions)

1. What are Negative Exponents?

Ans: A negative exponent is used when 1 is divided by repeated multiplication of a factor. Say, 1/n is given by n^{-1}, where -1 is the exponent. If a number is raised to negative exponents then it represented as the reciprocal of it. For example, 3 raised to -2 is represented by 3^{-2}, which is equal to 1/3^{-2}.

2. What do we get if the Exponent is 1 or 0?

Ans: If the exponent of a base number is one, then the value of the base remains unchanged. For example, 5^{1} = 5. If the exponent value is 0, then we get 1 as the final result. For example, 5^{0} = 1.

3. What Does 4 to the Power of 4 Mean?

Ans: If a number is said to be 'to the fourth power,' that just means that we need to multiply the number by itself four times. In this 4 to the power of 4 means, multiply 4 to the 4 times.