Laws of Exponents

An exponent refers to the number of times a number, called the base, is multiplied by itself.

Power is an expression that represents the multiplication of a number, called the base, by itself a certain number of times. The number of times the base is multiplied by itself is called the exponent.

In this article, we will cover:

• The Product of Powers with the Same Base
• The Quotient of Powers with the Same Base
• A Power to a Power
• The nth Root of a Power
• Zero Power
• FAQs

For example, in the expression 8^3, the base is 8 and the exponent is 3. This can be read as “8 to the power of 3” or “8 cubed.” The exponent tells us how many times the base is used as a factor in a multiplication problem. In this case, 8 is multiplied by itself 3 times, so the result is 8 x 8 x 8 = 512.

An exponent is a small number written above and to the right of the base. It is used to indicate the number of times the base is used as a factor in a multiplication problem. In example 8^3, the exponent 3 indicates that the base 8 is multiplied by itself 3 times, resulting in a value of 512.

So, while power is an expression that represents the multiplication of a number by itself a certain number of times, the exponent is the number of times the base is multiplied by itself.

Laws of Exponents

There are four main laws of exponents:

1. The Product of Powers with the Same Base

The Product of Powers with the Same Base states that if you have two or more powers with the same base, you can find their product by adding the exponents. This means that if you have two numbers that are both raised to a power, and the base of those two numbers is the same, you can multiply them by adding the exponents

a^m x a^n = a^(m+n) where m and n are real numbers.

Here are some examples to illustrate this law:

In these examples, we have two powers with the same base (2, 5, and x). To find the product of these powers, we add the exponents (3+4, 2+3, and 4+7). The result is the base raised to the sum of the exponents.

It’s important to note that this law only applies when the base of the powers is the same. If the bases are different, you cannot use this law to find the product.

2. The Quotient of Powers with the Same Base

The Quotient of Powers with the Same Base states that if you have two powers with the same base and you want to divide them, you can find the quotient by subtracting the exponents. This means that if you have two numbers that are both raised to a power, and the base of those two numbers is the same, you can divide them by subtracting the exponents.

a^m / a^n = a^(m – n) where m and n are integers and a is any non-zero term.

Here are some examples to illustrate this law:

In these examples, we have two powers with the same base (2, 5, and x). To find the quotient of these powers, we subtract the exponents (3-4, 7-3, and 9-4). The result is the base raised to the difference of the exponents.

It’s important to note that this law only applies when the base of the powers is the same. If the bases are different, you cannot use this law to find the quotient.

3. A Power to a Power

A Power to a Power states that if you have a power raised to another power, you can find the result by multiplying the exponents. This means that if you have a number that is raised to a power, and that power is itself raised to another power, you can find the result by multiplying the two exponents.

(a^m)^n = a^(m*n) where a is any non-zero term and m and n are integers.

Here are some examples to illustrate this law:

In these examples, we have a power that is raised to another power (2^3, 5^2, and x^4). To find the result, we multiply the exponents (3*4, 2*3, and 4*5). The result is the base raised to the product of the exponents.

It’s important to note that this law only applies when the base of the powers is the same. If the bases are different, you cannot use this law to find the result.

4. The nth Root of a Power

The nth Root of a Power states that if you have a power and you want to find its nth root, you can do so by dividing the exponent by n. This means that if you have a number that is raised to a power, and you want to find the root of that power, you can do so by dividing the exponent by the root you are looking for.

Here are some examples to illustrate this law:

Order of Operations FAQS

Grade 6 Science Worksheets

• Inquiry process
• Nature of Science
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• The Human Respiratory System
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Grade 6 Math Worksheets

• Probability
• Ratio
• Sequence & Patterns
• Division Word Problems  (New)
• Solving two step equations (New)
• One Step Equation  (New)
• Place Value  (New)
• Two Step Equation  (New)
• To convert percents to decimals and vice versa  (New)
• To convert Percents to Fractions and vice versa  (New)