Exponents and Powers

Exponents and powers are a combined simplification form of multiplication. By using exponent and power, we can solve multiplication easily. Power is an expression, which represents the repeated multiplication of a number. The factor, which is multiplied repeatedly, is called the base. The number of multiplications is called the exponent. For example, 5 is multiplied 3 times. So, 5* 5* 5 = 53, where 5 is base, 3 is exponent, and 53 is power. This expression 53 is known as 5 to the power of 3. Thus, you can take any multiplication factor to multiply the same factor multiple times and convert into power expression. After that, you will be able to identify the base and exponent from the expression.

General Form of Power and Exponent

Power and exponent are used to simplify the multiplication of the same factor. The power expression represents the whole process and the exponent defines the time of the factor to be multiplied. If the factor a, is multiplied for n times, the power expression is an.  Here, an is expressed as:

an = a* a* a* ……. * a (n times)

Hence, we can say that it is a simplified method of repeated multiplication.

Laws of Power and Exponent

The laws of exponents and powers are mainly based on the exponent part. Therefore, the laws are also known as the laws of exponents. The laws are defined below with expression.

1. Multiplication Law

This law defined that, when two exponents are multiplied with the same base factor, the two exponents are added in the result. The base or multiplication factor remains the same. The expression of multiplication law is:

am * an = am+n.

2. Division Law

As per division law, when two exponents are divided with the same base factor, the two exponents are subtracted in the result. The base will remain the same. The expression of this law is:

am / an = am-n.

3. Negative Exponent Law

As per negative exponent law, if any base has a negative exponent, it comes in the reciprocal with positive power to the base. The expression of negative exponent law is given below as:

a-n = 1/an

Rules of Exponents and Powers

There are several specific rules of exponents and powers. The rules are dependent on the laws of exponents. Here, we have given a brief explanation of the rules. Suppose, x and y are two integers and p and q are the exponents. Now, the rules are:

• If the power of any integer factor is o, the result will be one. The expression of this rule is x0 = 1. For example, 20 = 1.

• If the exponent of a power expression is an integer, the result exponent is the product of two exponents. The expression is, (xp)q = xpq. For example, (122)3 = 122*3 = 126.

• If the two different bases are multiplied with the same exponent, the result is the product of two bases and the exponent remains the same. The expression of this rule is xp * yp = (xy)p. For example, 22*32 = (2*3)2 = 62.

• If two different bases are divided with the same exponent, the result is the division of two bases and the exponent remains the same. The expression is, xp / yp = (x/y)p. For example, 22 / 32 = (2/3)2.

(Image will be uploaded soon)

Applications of Exponents and Powers

In the scientific field, these are the two most applied concepts. There are huge numerical calculations in different scientific fields. That is why exponents are necessary for scientific calculations. The necessity of powers and components can be explained with examples. The mass of the sun is measured as 1,989,000,000,000,000,000,000,000,000,000 Kg. The distance between the Earth and the Sun is measured to be 149,600,000 Km. These are huge numbers to memorize as well as to calculate. With the help of an exponent, these numbers can be broken into powers of 10. After using exponent, the distance between earth and Sun becomes 1.496 * 108 Km and the mass of the Sun becomes 1.989 * 1030 Kg.

Solved Examples

1. Identify the base, exponent, and power of the following expressions -

1. (16)2   b) (31)3

Solution:

1. The base is 16.

            The exponent is 2.

            The power is (16)2.

2. The base is 31.

            The exponent is 3.

            The power is (31)3.

2. Convert the multiplications into powers:

1. 5 × 5 × 5 × 5 × 5 × 5

2. 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3

Solution:

1. 5 × 5 × 5 × 5 × 5 × 5 = 56.

2. 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 38.