Index

In mathematics, an index (indices) is the power or exponent applied to a number or variable. For instance, the index of 2 is 4 in number 24. Indices is the plural term of index.
Constants and variables are used in algebra.

 A variable quantity, on the other hand, can be assigned any number or have its value altered.An index can be assigned to a number or a variable. 

A variable’s index (or constant) is a value that is raised to the variable’s power. Indices are also referred to as power. It displays how many times a given number must be multiplied.

Here, in the term xn,

• x is called the “base”
• n is called the “index”
• xn is read as x to the power of n.

Laws of indices

Indices are used to display how many times a number has been multiplied by itself. They can also be used to represent roots and fractions, such as the square root.

The principles of indices make it possible to alter expressions involving powers more quickly than if they were written out whole.
Product Law of indices

When multiplying expressions with the same bases, the product rule of exponents is employed. To multiply two equations with the same base, add the exponents while keeping the base the same, according to this formula.
Exponents with the same base are added using this rule. The rule can be used to simplify two expressions that share a multiplication operation.

23× 25= 2(3 + 5)= 28

Quotient Law of Indices

To reduce equations with the same bases, the quotient property of the index is used. To divide two equations with the same base, remove the exponents while keeping the same base, according to this principle.
This is useful for solving an equation without having to divide the numbers. The sole requirement is that the terms have the same base. 

25/23= 25 – 3= 22

Zero Law of Exponents

When the indices of an expression is 0, the zero property of index is used. Any number (other than 0) raised to 0 equals one, according to this characteristic.
It’s worth noting that 00 isn’t specified.
This will show us that the value of a zero exponent is always equal to 1 regardless of the base.

20= 1 

Negative Law of Indices

When an index has a negative number, the negative feature of exponents is employed. This characteristic states that the reciprocal should be used to transform any negative exponent to a positive exponent.
With the change in sign of the exponent values, the expression is shifted from the numerator to the denominator.

2-2= 1/(2)2

Power of a Power Law of Exponents

To simplify calculations of the form (am )n , the power of a power property of index is employed. According to this principle,When we have a single base with two exponents, we simply multiply the Indices.
The two exponents are alternately available. 

To make a single exponent, simply multiply these numbers. 

(22)3= 26

Power of Product Rule of Indices

To find the outcome of a product raised to an index, use the ‘power of a product property of exponents’.
Distribute the exponent to each multiplicand of the product, according to this characteristic.

(xy)3= x 3 .y3

Power of a Quotient Rule of Index

To find the outcome of a quotient raised to an Index, the power of a quotient property of exponents is applied.
Distribute the exponent to both the numerator and the denominator, according to this characteristic.

(x/y)3= x3/y3

Examples:

• -2 × -2 × -2 = (-2)3
• a × a × a × a × a × a = a6

Exponents are important because, without them, when a number is repeated by itself many times it is very difficult to write the product. 

For example, it is very easy to write 57 instead of writing 5 × 5 × 5 × 5 × 5 × 5 × 5.

Types of Exponents

Negative Exponents

A negative exponent indicates how many times the reciprocal of the base must be 1/an multiplied. For instance, if a-n is supplied, it can be expanded to.
It indicates we have to multiply 1/a ,’n’ times the reciprocal of a.
When writing fractions using exponents, negative exponents are used. 

Exponents with Fractions

A fractional exponent is when the index of a number is a fraction.
Fractional exponents include square roots, cube roots, and the nth root. The square root of the base is a number having a power of 1/2.
Similarly, the cube root of the base is an integer with a power of 1/3.

e.g  52/3,  105/6, etc

Decimal Exponents

A decimal exponent is one in which the exponent of a number is expressed in decimal form. Because evaluating the correct answer of any decimal exponent is a little challenging, we use an approximation in such circumstances.
To solve decimal exponents, convert the decimal to fraction form first.

For example, 41.5 can be rewritten 43/2after simplification 8.

Conclusion

In mathematics, indices are a useful tool for denoting the process of raising or lowering a number to a power or root. Taking a power is just the process of multiplying a number by itself multiple times, whereas taking a root is the same as taking a fractional power of the number.