Exponents

An exponent is a mathematical notation that represents how many times a number, called the base, is multiplied by itself. 

For example, in 5 × 5 × 5, 5 is multiplied 3 times. So, in exponent form, it is written as 5³, where 5 is the base.

The number written above 5 in superscript (i.e., 3) is called the exponent. Often called the index (plural: indices) or power, it represents how many times the number is repeated in multiplication, known as exponentiation.

The above expression 5³ is read as ‘5 to the power 3,’ or ‘5 to the 3rd,’ or ‘5 cubed.’ 

Here are a few more examples of exponents:

• 2⁵ = 2 × 2 × 2 × 2 × 2 = 32
• 3² = 3 × 3 = 9
• 6³ = 6 × 6 × 6 = 216

In general, if x is an integer raised to a positive integer n, then it is expressed as xⁿ

Symbol

The symbol ^ is also used to represent exponents while typing. This is called the ‘carrot.’ 

Rules

Some important laws that are used while solving expressions involving exponents are listed:

Properties	Mathematical Form
Product of Powers	xm × xn = xm + n
Quotient of Powers	${\dfrac{x^{m}}{x^{n}}=x^{m-n}}$
Power of a Power	(xm)n = xmn
Power of a Product	(xy)m = xmym
Power of a Quotient	${\dfrac{x^{m}}{y^{m}}=\left( \dfrac{x}{y}\right) ^{m}}$
Zero Exponent	x0 = 1
One Exponent	x1 = x
Negative Exponent	${x^{-m}=\left( \dfrac{1}{x}\right) ^{m}}$
Fractional Exponent	${x^{\dfrac{m}{n}}}$ = ${\left( \sqrt[n]{x}\right) ^{m}}$ or ${\sqrt[n]{x^{m}}}$

These rules make it easier to perform operations with exponents without needing to manually multiply the base repeatedly. 

Special Cases

Negative Exponents

When the exponent of a number is negative, it is called a negative exponent. 

While a positive exponent tells us how many times to multiply a base by itself, a negative exponent tells us to take the reciprocal (or ‘flip’) of the base raised to its corresponding positive exponent. Thus, a negative exponent inverts the base, shifting it from the numerator to the denominator (or vice versa) of a fraction.

For example,

2^-3 can be expressed as ${\dfrac{1}{2^{3}}}$ 

Decimal Exponents

When the exponent of a number is in decimals, it is called a decimal exponent. It is written as a combination of integer powers and roots. 

For example,

25^0.5 is broken down as
= ${25^{\dfrac{1}{2}}}$, where 25^0.5 represents the square root of the base 25 (i.e., ${\sqrt{25}}$)

= 5

Sometimes, evaluating decimal exponents can be challenging. In such cases, we find the approximate value.

For example, 

5^0.5 = ${5^{\dfrac{1}{2}}}$ ≈ 2.236

In Scientific Notations

Exponents are often used in scientific notation to represent very large or very small numbers using powers of 10. In this system, a positive exponent is used for numbers greater than 1, while a negative exponent is used for numbers less than 1.

To express a number in scientific notation, we follow the given steps:

1. Placing the decimal point after the first non-zero digit in the number.
2. Multiplying by 10 raised to the power that represents how many places the decimal point is shifted

For large numbers, the exponent is positive, indicating how many times the decimal point is shifted to the left. For small numbers, the exponent is negative, showing the number of times the decimal point is shifted to the right.

Let us express 7600000000000 and 0.00000003874 in scientific notation form.

7600000000000 = 7.6 × 10¹²

0.00000003874 = 3.874 × 10^-8 

To have a better understanding of the use of exponents in writing numbers in scientific notation, click here.