Exponent Rules

1. Product Rule: When multiplying powers with the same base, add the exponents.
a^m × aⁿ = a^m+n

2. Quotient Rule: When dividing powers with the same base, subtract the exponents.

a^m/aⁿ = a^m-n 

3. Power of a Power Rule: When raising a power to another power, multiply the exponents.

(a^m)ⁿ = a^m·n

4. Power of a Product Rule: Distribute the exponent to each factor inside the parentheses.
 (ab)ⁿ = aⁿ. bⁿ

5. Power of the Quotient Rule: Distribute the exponent to both the numerator and denominator.

(a/b)ⁿ = aⁿ/bⁿ as long as b ≠ 0

6. Zero Exponent Rule: Any non-zero base raised to the power of 0 is 1.
a⁰ = 1 as long as a ≠ 0

7. Negative Exponent Rule: A negative exponent means to take the reciprocal.

a^-n  = 1/aⁿ, a ≠ 0

1. Product of Powers: When you multiply expressions with the same base, add the exponents. If a^m . a^h is the expression, then the product will be:  am. ah = am+n

Example: x³.  x² = x³⁺² = x⁵

2. Quotient of Powers: When you divide expressions with the same base, subtract the exponents, like a^m/aⁿ = a^m-n as long as α = 0
Example: y⁵/y² = y^5-2 = y³

3. Power of a Power: When you raise an exponent to another exponent, multiply them, like
(a^m)ⁿ =   a^{m . n}

Example: (x²)⁴ = x^{2 . 4} = x²

4. Power of a Product: Distribute the exponent to each factor inside the parentheses, like 
(a b)ⁿ= aⁿ . b²

Example: (3x)²= 3² . x² = 9x²

5. Power of a Quotient: Distribute the exponent to both the numerator and the denominator, 
like  (a/b)ⁿ = aⁿ/bⁿ as long as b=0

Example: (x/y)³ = x³/y³

6. Zero Exponent: Anything (except 0) raised to the 0 power will be 1 
a^o= 1 (if α=0)

Examples: 5o = 1

7. Negative Exponent: A negative exponent means to take the reciprocal, like
a^-n = 1an (if α = 0)

Example: x^-3 = 1/x³

If you're dividing two powers that have the same base, you just subtract the exponents, like 

a^m/aⁿ = a^m-n as long as =0

Example: x⁷/x³ = x^7-3 = x⁴

The Zero Law of Exponents (also called the Zero Exponent Rule) says that any non-zero number raised to the power of 0 is equal to 1.
a^o = 1 (as long as α=0)

If you raise any non-zero number to the 0 power, the result will always be 1, regardless of how small or big the number is. 

If a number has a negative exponent, take the reciprocal to make the exponent positive.

a^-n = 1/aⁿ , a ≠ 0

The Power of a Power Law, also called the Power Rule, tells you what to do when you raise a power to another power.

(a^m)ⁿ = a^{m . n}

If you have an exponent raised to another exponent, you multiply the exponents.

The Power of a Product Rule helps you simplify expressions when a product (multiplication) is raised to a power.

(abⁿ)ⁿ = aⁿ . bⁿ

When you have two things multiplied together inside parentheses, and the whole thing is raised to a power, we apply the exponent to each factor.

The Power of a Quotient Rule helps you simplify expressions when a fraction (quotient) is raised to a power.

(a/b)ⁿ = aⁿ/bⁿ as long as α = 0 and b ≠ 0

When a fraction is raised to a power, you give that power to both the top and the bottom of the fraction.

The fractional exponents rule (also called the rational exponents rule) helps you understand what it means when an exponent is a fraction.

a^m/n = (n√a)  ^m = n√a^m, where a ≥ 0

The denominator of the fraction (the bottom number) tells you the n^th root. The numerator (the top number) tells you the power.

The Exponent rules are applied when problems with exponents are included. They are applied in real life, such as population change, money growth, and so on. Given below are a few real-life applications of the topic given.  Are some exponent rules that we use in our lives.

• Banking and Investments: When you keep money in the bank or invest it, the value increases over time. The longer it stays, the faster it grows—this is called compound interest.

• Population and Bacteria Growth: Populations of animals, humans, or bacteria can grow very fast. Scientists use exponents to predict how big the population will get.

• Measuring Earthquakes: Each time the earthquake scale increases by one point, the strength of the quake is 10 times greater. A magnitude 6 is much more powerful than a magnitude of 5.

• Computer Storage and Memory: Digital storage (like kilobytes, megabytes, gigabytes) is based on multiplying numbers quickly. For example, your phone might have 64 GB of storage holds a very large amount of data when written in full.

• Radiation and Medicine: In treatments like cancer therapy, doctors use formulas involving exponents to measure how much radiation should be given to the patient.