# 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!