# An Introduction to Exponents

The mass of the moon is 7,350,000,000,000,000,000,000,000 kg. Can you read this number? It’s not so easy to read or even recognize all these long digits accurately. Therefore in order to give a precise evaluation of the mass of the moon, we can use Exponents.

In mathematics, **natural numbers** are defined as the number which is a set of all counting numbers starting from \(1\). The natural number includes all the positive Integers (from 0 to \(\infty\)). Fractions are not a part of natural numbers.

An expression that consists of a repeated power of multiplication of the same factor is called as Power/Exponent/Indices.

Consider an example like \(5^{2}\), the number 5 is called the base, whereas 2 is the power/indices/exponent of the expression.

The value of the expression is derived by multiplying the base as many time as the number of power. In the example above, the power is 2, thus the value becomes \(5 \times 5 = 25\).

**Example**- Express the number 245,000,000 into the exponent form.

Solution- The term can be written in the form of \(2.45 \times 10^{8}\).

**Example**- What is the exponential form of 8 in the expression given below-

\(\large \mathbf{2^{3}}\)

Solution- The exponential form is the power in the expression. In the given expression the power of the base is 3. Hence 3 is the Index/ Power.

Also, the value of the expression equals to 8.

### Negative power

The negative power is almost similar to the positive power of the exponent. The only change in the negative power is that the value of the expression is the reciprocal of the value obtained in the positive case.

Take an Example: \(\large 5^{-2}\)

The negative power can also be written as- \(\large \frac{1}{5^{2}} = \frac{1}{25}\).

Let us have a look at few more examples-

**Example**- Find the value of the given expression (in fractional form)

(i) \(\large 3^{-4}\)

(ii) \(\large 8^{-3}\)

(iii) \(\large 2^{-6}\)

Solution-

(i) \(\large 3^{-4}\)

The expression can be written as-

\(\large \frac{1}{3^{4}} = \frac{1}{81} \)

(ii) \(\large 8^{-3}\)

The expression can be written as-

\(\large \frac{1}{8^{3}} = \frac{1}{512} \)

(iii) \(\large 2^{-6}\)

The expression can be written as-

\(\large \frac{1}{2^{6}} = \frac{1}{64}\)

Relation between positive and the negative power-

Let us consider a base number to be a, and the power to be x,

The relation between the positive and the negative expression is given as-

\(\large \mathbf{\large a^{x} = \frac{1}{a^{-x}}}\)

Value of expression when the base value is 0 or 1:

0 and 1 are the only power that gives the same value, i.e. 0 and 1 respectively, for different exponent values.

Such as \(\large 0^{13} = 0^{14341} = 0\)

\(\large 1^{3} = 1^{1432} = 1\)

Negative power of 0 and 1-

The negative power of 0 does not exist in nature. As the negative power would lead to making the denominator to be 0, therefore the negative power of 0 cannot exist in nature.

Whereas the negative power of 1 exist in nature and is equal to the positive power of 1, as it would also result in the value to be equal to 1.

In General form, we can write as-

\(\mathbf{\large 0^{a} = 0}\) (where ‘a’ is a positive number)

\(\large \large 1^{a} = 1\) ( where ‘a’ can take positive as well as negative value both)

When the power is 0-

When the exponent value is equal to 0, the value of the expression is always equal to 1.

What about \(\mathbf{\large 0^{0}}\)–

Based on the explanation stated above the value of the expression can either take 0 or 1.

But the value of such an expression cannot take 2 values at a time. Therefore the correct value for \(\large 0^{0}\) is equal to 1.

\(\large \mathbf{0^{0} = 1}\)

**Example:** What is the exponential form of \(256\)? Mention its base and exponent.

Solution: \( 256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \) = \(2^{8}\)

In the above example, “2” is the base which is in 8 times repeated multiplication, equals “\( 256 \)”.

So, the base is “2” and the exponent is “8”.

**Example:** Which one is smaller \(5^{3}\) or \(3^{5}\)?

Solution:

First we need to calculate the value of the given expression.

\(5^{3}\) = \(5 \times 5 \times 5 = 125 \)

and, \(3^{5}\) = \(3 \times 3 \times 3 \times 3 \times 3\) = \(243\)

Thus \(5^{3} = 125\) and \(3^{5} = 243\)

Therefore, \(5^{3} < 3^{5}\)