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.

Laws of Indices

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. Indexes are also referred to as powers or exponents.

xⁿ=x×x×x×x×………×xn times

Before we begin working with indexes, we must first understand certain fundamental principles or laws that govern them. These rules are applied while conducting algebraic operations on indices and solving algebraic equations, which includes it. Let’s have a look at the various methods for calculating with indices.

1. If the index of a constant or variable is ‘0,’ the outcome will be one, regardless of the base value.>X⁰=1 for any value of x
2. If the index has a negative value, the reciprocal of the positive index raised to the same variable might be used.>X^-r= 1 / X^r
3. We must sum the powers of two variables with the same base and raise them to that base to multiply them.>X^r.X^s=X^r+s
4. To divide two variables with the same base, subtract the denominator’s power from numerator’s power and raise it to that base.>X^r/ X^s=X^r-s
5. When a variable with one index is raised with another index, both indices are multiplied together and raised to the same base power.>Xrs=X^rs
6. When two variables with different bases but the same indices are multiplied, the base must be multiplied and the same index must be raised to the multiplied variables.>X^r.Y^r=XY^r
7. When two variables with different bases but the same indices are divided, the bases must be divided and the same index must be raised.>X^{r /} Y^r=XY^r
8. The radical form can be used to indicate an index in the form of a fraction.>X ^{r /s}=s √X^r

What are Indices

The little floating number that appears after a number or letter is called an index or power. Indexes is the plural form of index. The number of times a number or character has been multiplied by itself is represented by an index.

• . X²(Read as ‘X squared’) means X×X. X  has been multiplied by itself twice. The index, or power, here is 2.
• . X³(Read as ‘X cubed’) means X×X×X.  X has been multiplied by itself three times.
•  .X⁴ (read as ‘X to the power of 4′) means X×X×X×X. X has been multiplied by itself four times, and so on.

We must be able to employ the laws of indices in a number of ways in order to calculate with indices.

Types of Indices

To understand the types of problems we’re going to face in order to learn to solve indices we’re going to see some solved examples and will observe how its solved.

Example: Given that 10^0.48 = x, 10^0.70 = y and x^z = y², then the value of z is close to:

 x^z=y²

 ⟹10^0.48z = 10^2×0.70 = 10^1.40

 ⟹z=140 / 48=35 / 12 = 2.9

Example: The value of 10¹⁵⁰ ÷ 10¹⁴⁶

 (10)¹⁵⁰ ÷ (10)¹⁴⁶ = 10¹⁵⁰/ 10¹⁴⁶ 

 = 10^150-146

 = 10⁴ 

 =10000