Lesson Objectives
• Demonstrate an understanding of exponents
• Learn how to use the product rule for exponents
• Learn how to use the quotient rule for exponents
• Learn how to use the power rules for exponents
• Learn how to simplify with negative exponents and the power of zero

## Rules of Exponents

At this point, we should understand the basic idea of an exponent. Exponents allow us to conveniently write the repeated multiplication of the same number in a more compact form. As an example:
3 • 3 • 3 • 3 = 34
In our expression 34, the 3 or larger number is known as the base and the 4 or smaller number is known as the exponent. In this case, 34, read as "3 to the 4th power" tells us we have four factors of the base 3. A whole number exponent larger than 1 represents the number of factors of the base.
59 » Tells us we have 9 factors of our base 5
1119 » Tells us we have 19 factors of our base 11
Before we start working with polynomials, it is important to have a good understanding of the various rules that are used when working with exponents. In this lesson, we will review the product rule for exponents, the quotient rule for exponents, negative exponents, the power rules for exponents, and the exponent of zero.

### Product Rule for Exponents

When we multiply exponential expressions with the same base, we keep the base the same and add exponents.
am • an = am + n
54 • 53 = 5(4 + 3) = 57
We keep our base 5 the same and add exponents (4 + 3 = 7).
x3 • x12 = x(3 + 12) = x15
We keep our base x the same and add exponents (3 + 12 = 15).

### Quotient Rule for Exponents

When we divide exponential expressions with the same base, we keep the base the same and subtract the exponent in the denominator from the exponent in the numerator. $$\frac{a^m}{a^n}=a^{m - n}\hspace{.25em}$$ $$\frac{8^9}{8^4}=8^{(9-4)}=8^5$$ We keep our base 8 the same and subtract exponents (9 - 4 = 5). $$\frac{x^{14}}{x^3}=x^{(14\hspace{.1em}- \hspace{.1em}3)}=x^{11}$$ We keep our base x the same and subtract exponents (14 - 3 = 11).

### The Power of Zero

Any non-zero number raised to the power of zero is 1.
a0 =  1
6220 = 1
Raising 622 to the power of 0 results in 1.
32550 = 1
Raising 3255 to the power of 0 results in 1.

### Negative Exponents

When we raise a number to a negative power, we take the reciprocal of the base and make the exponent positive. $$a^{-n}=\frac{1}{a^n}$$ An easier way to think about this rule is to remember when we drag a number in exponent form across a fraction bar, we keep the base the same and change the sign of the exponent. $$a^{-n}=\frac{a^{-n}}{1}=\frac{1}{a^n}$$ $$\frac{a^{-n}}{b^{-m}}=\frac{b^m}{a^n}$$ $$7^{-4}=\frac{1}{7^4}$$ We take the reciprocal of 7 and make the exponent (-4) positive. $$\frac{x^{-2}}{y^{-9}}=\frac{y^9}{x^2}$$ We put x into the denominator and change the exponent to positive. We put y in the numerator and change the exponent to positive.

### Power to Power Rule for Exponents

When we raise a power to another power, we keep the base the same and multiply the exponents.
(am)n = am • n
(92)3 = 92 • 3 = 96
We keep our base 9 the same and multiply the exponents (2 • 3 = 6).
(x8)11 = x8 • 11 = x88
We keep our base x the same and multiply the exponents (8 • 11 = 88).

### Raising a Product to a Power

We can raise a product to a power by raising each factor to the power.
(ab)m = am • bm
(2x)3 = 23x3
We raise both factors, 2, and x to the power of 3.
(xy)19 = x19y19
We raise both factors, x, and y to the power of 19.

### Raising a Quotient to a Power

We can raise a quotient to a power by raising the numerator and denominator to the power. $$\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}$$ $$\left(\frac{x}{y}\right)^4=\frac{x^4}{y^4}$$ We raise the numerator, x, and the denominator, y, to the power of 4. $$\left(\frac{4}{x}\right)^{13}=\frac{4^{13}}{x^{13}}$$ We raise the numerator, 4, and the denominator, x, to the power of 13.
Let's take a look at a few examples that involve the rules of exponents.
Example 1: Simplify each. $$\frac{(2x^5y^5)^2(2x^{-3}y^{-1})^4}{(2x^9y^7)^0 (2^3x^4y)^2}$$ Let's begin by working on the numerator: $$(2x^5y^5)^2(2x^{-3}y^{-1})^4$$ $$(2^2x^{10}y^{10})(2^4x^{-12}y^{-4})$$ $$2^6x^{-2}y^{6}$$ Now, let's replace our numerator in the original problem: $$\frac{2^6x^{-2}y^{6}}{(2x^9y^7)^0 (2^3x^4y)^2}$$ Now, let's work on our denominator: $$(2x^9y^7)^0 (2^3x^4y)^2$$ $$1(2^3x^4y)^2$$ $$2^6x^8y^2$$ Now, let's replace our denominator: $$\frac{2^6x^{-2}y^{6}}{2^6x^8y^2}$$ $$\require{cancel}\frac{\cancel{2^6}x^{-2}y^{6}}{\cancel{2^6}x^8y^2}$$ $$x^{-10}y^{4}$$ $$\frac{y^4}{x^{10}}$$ Example 2: Simplify each. $$\frac{(xy)^{-1}(x^4y^3z^5)^{-3}}{(x^5y^9z^{11})^{-2}}$$ Let's begin by working on the numerator: $$(xy)^{-1}(x^4y^3z^5)^{-3}$$ $$(x^{-1}y^{-1})(x^4y^3z^5)^{-3}$$ $$(x^{-1}y^{-1})(x^{-12}y^{-9}z^{-15})$$ $$x^{-13}y^{-10}z^{-15}$$ Now, let's replace our numerator in the original problem: $$\frac{x^{-13}y^{-10}z^{-15}}{(x^5y^9z^{11})^{-2}}$$ Now, let's work on our denominator: $$(x^5y^9z^{11})^{-2}$$ $$x^{-10}y^{-18}z^{-22}$$ Now, let's replace our denominator: $$\frac{x^{-13}y^{-10}z^{-15}}{x^{-10}y^{-18}z^{-22}}$$ $$x^{-3}y^{8}z^7$$ $$\frac{y^{8}z^7}{x^3}$$

#### Skills Check:

Example #1

Simplify each. $$\frac{2zx^{3}y^{-3}}{zx^{2}\cdot \left(2y^{0}z^{-4}\right)^{0}\cdot \left(x^{2}y^{-1}z^{-3}\right)^{-4}}$$

A
$$\frac{2x^{9}}{y^{7}z^{12}}$$
B
$$\frac{1}{x^{2}y^{2}z}$$
C
$$\frac{y^{21}}{z^{15}x^{9}}$$
D
$$\frac{x^{6}}{4y^{6}z^{2}}$$
E
$$\frac{z^{6}}{8}$$

Example #2

Simplify each. $$\frac{zx^{4}y^{-4}\cdot \left(x^{2}y^{-2}z^{-1}\right)^{-3}}{y^{3}z^{4}}$$

A
$$\frac{1}{xy^{3}}$$
B
$$1$$
C
$$\frac{y^{8}}{z^{36}}$$
D
$$\frac{1}{x^{2}y}$$
E
$$\frac{4y^{2}z^{6}}{x^{10}}$$

Example #3

Simplify each. $$\frac{\left(2pq^{4}r^{-2}\right)^{-1}\cdot \left(2q^{4}\right)^{4}}{r^{3}}$$

A
$$1$$
B
$$\frac{q^{16}}{p^{11}r^{5}}$$
C
$$\frac{8q^{12}}{pr}$$
D
$$\frac{q^{24}}{16p^{20}r^{36}}$$
E
$$8p^{7}r^{14}q^{22}$$