# What are the Laws of Exponents?

## Properties of Exponents

Hi, and welcome to this video on the properties of exponents!

Working with polynomial, radical, and rational functions often times require us to perform algebraic operations with powers. Recall that a power is nothing more than a base that is raised to an exponent. So let’s take a look at the Properties of Exponents that are used to simplify algebraic expressions with powers.

The **Product of Powers** property applies to powers with the same base. When asked to multiply powers with the **same base**, simply add the exponents. For example,

This rule makes intuitive sense if you expand each power like this:

(b • b)(b • b • b) counting up the bases of “b” that are being multiplied results in 1,2 3, 4, 5. \(b^5\).

Therefore, the general form of the Product of Powers rule is \(b^mb^n = b^{(m+n)} \)

The **Quotient of Powers** property also applies to powers with the same base; however, the rule requires the subtraction of exponents. Let’s look at an example:

\(\frac{b^5}{b^3}\) in expanded form would look like this: \(\frac{\text{b • b • b • b}}{\text{b • b • b • b}}\)

Canceling out common factors of “b” from the numerator and the denominator would simplify to be \(b^2\).

The **Quotient of Powers** property allows a quicker, more efficient result. Simply subtract the exponent of the denominator from the exponent of the numerator: 5 – 3 = 2.

The general form for Quotient of Powers property is \(\frac{b^m}{b^n} = b^{(m-n)}\).

The **Power of a Power** property allows us to raise a power to another exponent. Once again, expanding an expression makes understanding the rule a bit easier. Suppose the power \(b^3\) is squared:

\((b^3)^2\) in expanded form it would look like this (b • b • b)(b • b • b). Using the Product of Powers property, or simply counting up the bases of b, results in \(b^6\).

The general form of this property is: \((b^m)^n=b^{mn}\).

Expanding on the Power of a Power property results in additional tools that are used to simplify expressions with powers.

For example, suppose the base of a power is a monomial with a coefficient and a variable.

\((3b^4)^2\) expands to \((3b^4)(3b^4)\). Rearranging the factors to group the coefficients and the powers with the same base, b, and simplifying gives us \((3 \times 3 \times b^4 \times b^4)=9b^8\)

This is called the **Power of a Product** property, and illustrates that the exponent of each factor in the base must be multiplied by the power outside the parentheses.

The general form for the **Power of Product** property is: \((b \times c)^x=b^x \times c^x\)

The **Power of Quotient** property is similar in that the exponent of a rational base is multiplied by the exponents in both the numerator and denominator. Let’s look at some examples:

\((\frac{b^3}{c^2})^2\) expands to \((\frac{b^3}{c^2})(\frac{b^3}{c^2})\).

The Product of Powers property results in the simplified expression, b6c4.

However, applying the Power of Quotient property directly is often more efficient:

\((\frac{b^3}{c^2})^2 = \frac{b^{3 \times 2}}{c^{2 \times 2}} = \frac{b^6}{c^4}\)Here’s another example:

\((\frac{2b}{c^2d})^3\) expands to \((\frac{2b}{c^2d})(\frac{2b}{c^2d})(\frac{2b}{c^2d})\).

Rearranging factors and applying the Product of Powers property results in:

If we were to apply the Power of Quotient property, it would look like this:

\((\frac{2^{1 \times 3}b^{1 \times 3}}{c^{2 \times 3}d^{1 \times 3}}) = \frac{2^3b^3}{c^6d^3} = \frac{8b^3}{c^6d^3}\)The general rule for the Power of Quotient property is: \((\frac{b}{c})^x = \frac{b^x}{c^x}\)

As you can see, these rules are essential in simplifying expressions within our work with various algebraic functions.

Thanks for watching this review of exponents! Happy studying!