Exponents

An exponent is a number that indicates how many times a base should be multiplied by itself. Exponents help represent large numbers easily. In the figure given below, we get to see an example of an exponent and base.

The term xn here means,

x is known as the base

n is known as an exponent

xn is read as ‘ x to power n’

n times product exponent formula: \(x.x.x.x … n times = xn\)

Multiplication Rule: \(xm . xn = x(m + n)\)

Division Rule: \(\frac{x^m}{x^n} = x^{m-n} \)

Power of the product rule: \((xy)^n = x^n \cdot y^n \)

Power of a fraction rule: \(\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n} \)

Power of the power rule: \(\left(x^m\right)^n = x^{mn} \)

Zero Exponent: \((x)0 = 1\), if x 0

One Exponent: \(x^1 = x \)

Negative Exponent: \(x^{-n} = \frac{1}{x^n} \)

Fractional Exponent: \(x^{m/n} = \sqrt[n]{x^m} = \left(\sqrt[n]{x}\right)^m \)= \(\sqrt[n]{x^m} = (\sqrt[n]{x})^m\)

There are seven laws of exponents, and below they are explained in detail:

• Multiplication Law: If two exponential terms with the same base are multiplied, retain the base and add the exponents.

Example: \(a^3 \cdot a^5 = a^{3+5} = a^8 \)

• Division Law: When dividing exponential terms with the same base, keep the base and subtract the exponents.

 Example: \(\frac{45}{42} = 45 - 2 = 43 \)

• Power of a Power Rule: When an exponential term is raised to another power, multiply the exponents.

 Example: (\((a^3)^4 = a^{3 \cdot 4} = a^{12} \) 

• Power of a Product Rule: When two terms with same power and different bases are multiplied, the bases are multiplied and the power remains the same.  

Example: \(2² × 4² = 4 × 16 = 64\)

• Power of Quotient Rule: If two terms with different bases and same power are divided, then the answer will have the same power, but the base will be the quotient that we get when two bases are divided.

Example: \( 8² ÷ 2² = (8 ÷ 2)² = 4² = 16\)

Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1.

 Example: \(51^0 = 1 \)

Negative Exponent Rule: When the exponent is negative, we can convert the base into its reciprocal to make the exponent positive. 

Example: \(4^{-2} = \left(\frac{1}{4}\right)^2 = \frac{1}{4^2} = \frac{1}{16} \)
 

A negative exponent indicates the power of the reciprocal of the base. To simplify, take the reciprocal of the base and then apply the positive version of the exponent using standard rules. This can be represented as: 

                            \(x^{-n} = \left(\frac{1}{x}\right)^n \)

For example: \(x^{-n} = \left(\frac{1}{x}\right)^n \)

What are Decimal Exponents? 

A decimal exponent is another term for a fraction exponent. If an exponent is in the decimal form, then we should change it into a fraction form to solve it easily. Given below is an example for better understanding.
Simplify 61.5

Solution: We can replace 1.5 as \(\frac{3}{2}\)

        \(6^{1.5} = 6^{3/2} = (\sqrt{6})^3 \approx 14.7 \)

What are Exponents with Fractions? 

Exponents that are fractions are also known as radicals. These fractional powers represent roots,
such as square roots, cube roots, and the general nth root. 
A fractional exponent is expressed in the form: \(a^{m/n} \)

This signifies, \(a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m} \)
Where,
a is the base
m is the power to which the base is raised 
n is the index of the root (the denominator of the fraction)
For example: \(8^{1/3} = \sqrt[3]{8} = 2 \).

What is Scientific Notation of Exponents? 

Scientific notation is a method of expressing large numbers conveniently using the powers of ten. It follows a specific format, which is, a10n. Here, a is a number between 1 and 10 and n can either be a positive or negative exponent. For, e.g., 10,000 can be written as \(1 × 10⁴\). Similarly, 0.01 can be written as \(1 × 10⁻²\).

Exponents can often be a confusing topic for students. Knowing these few tips will help students tackle issues efficiently.

• Memorize the basic rules like \(a^m \cdot a^n = a^{m+n} \quad \text{and} \quad (a^m)^n = a^{mn} \)

• Always rewrite negative exponents as fractions for clarity,

• Always break complex expressions into smaller steps before solving.

• Convert roots to fractional exponents to make calculations easier.
   
• Substitute small numbers to verify the result and avoid errors in powers.

We can find exponents all around us. When we have to express a very large or small number, we use exponents.
Here's a look at some of their real-life applications:

• Finance: Exponents are used to calculate and measure how investments grow over a period of time. For e.g., compound interest is calculated using the formula \(A = P \left( 1 + \frac{r}{n} \right)^{nt} \) where nt is an exponent. 
   
• Sound Intensity: Exponents are fundamental to logarithmic scales that measure sound intensity.
   
• Astronomy and Light Years: It is used to measure the huge distances between galaxies expressed using large numbers, often involving exponents of 10.
   
• Biology: In the field of biology, it is used to measure the growth of population. For example, exponents play an important role while calculating the rate at which a colony of virus multiplies. 
   
• Describing Complex Patterns, Fractal Geometry, and Nature: The self-similar and scaling properties of intricate natural patterns like snowflakes and coastlines are mathematically described using exponents.