# Exponents

## Introduction

Exponent notation is a quick way to indicate a number multiplies itself by itself a number of times. Rather than writing: $6\times6\times6\times6\times6\times6\times6\times6$ it’s much more efficient to use the exponent format of: $6^{8}$.

The proper use of exponents can help students avoid errors when solving mathematical equations and simplifing algebraic expressions.

## Properties of Exponents

Although there are many exponent properties, understanding and learning how each property is applied for use with numbers is essential for the mastery of operations with exponents.

### Zero Exponent Property

Any number that is raised to the zero power is equal to 1.

Example:

\begin{align} a^{0}&= 1\\ 1,\!000,\!000^{0}&=1 \end{align}

### Negative Exponent Property

Negative exponents are written as a fraction with 1 in the numerator and the number raised to the positive power in denominator.

Example:

\begin{align} a^{-b}&= \frac{1}{a^{b}}\\ 6^{-2}&=\frac{1}{6^{2}}\\ \frac{1}{6^{2}}&=\frac{1}{36} \end{align}

### Product of Powers Property

To multiply the same bases raised to powers, add the powers.

Example:

\begin{align} a^{b}\times a^{c}&= {a^{b+c}}\\ 4^{2} \times 4^{3} &=4^{5}\\ 16\times64&=4^{5}\\ 4^{5}&=1024 \end{align}

### Quotient of Powers Property

When dividing same bases raised to powers, subtract the powers.

Example:

\begin{align} a^{b}\div a^{c}&= {a^{b-c}}\\ 5^{3} \div 5^{2} &=5^{3-2}\\ 5^{3} \div 5^{2} &=5^{1}\\ 125\div 25&=5^{1}\\ 5^{1}&=5 \end{align}

### Power of a Product Property

To simplify the product of different bases raised to the same power, multiply the bases and raise to the power indicated.

Example:

\begin{align} a^{c}\times b^{c}&=(ab)^{c}\\ 3^{3}\times2^{3}&= (3\times2)^{3}\\ 27\times8&=6^{3}\\ 216&=6^{3}\\ 6^{3}&=216 \end{align}

### Power of a Quotient Property

If unlike numerators and denominators are each raised to the same power, the fraction can be raised to the indicated power.

Example:

\begin{align} \frac{a^{c}}{b^{c}}&= {(\frac{a}{b})^{c}}\\ \frac{3^{2}}{4^{2}}&= {(\frac{3}{4})^{2}}\\ \frac{9}{16}&={(\frac{3}{4})^{2}}\\ {(\frac{3}{4})^{2}}&={\frac{9}{16}} \end{align}

### Power of a Power Property

For a number raised to a power then raised to another power, raise the base to the product of the two exponents.

Example:

\begin{align} (a^{b})^{c} &= a^{bc}\\ (2^{3})^{2}&=2^{6}\\ 8^{2}&=2^{6}\\ 2^{6}&=64\\ \end{align}

## Scientific Notation

Mathematicians devised scientific notation in order to manage very large and very small numbers.

To write scientific notation, move the decimal point to make a number in the ones place then multiply times ten to the power of the number of place values you moved the decimal point. For a very large numbers, the exponent will be positive, and for a number less than one, the exponent will be negative.

Example:

\begin{align} 1,\!867,\!000,\!000.&=1.867 \times 10^{9}\\ 0.000678&=6.78 \times 10^{-4} \end{align}

## Exponential Growth and Decay

When a number grows or deceases exponentially, the change is rapid.

Growth: $y=a(1+r)^{x}$
Decay: $y=a(1-r)^{x}$

a = base amount
r = rate
x = time

It’s amazing how quickly things grow or decay exponentially.

Example: 1,000 people watched a video online. The video went viral, and the number of viewers doubled every day. How many viewers were there after two weeks?

\begin{align} y&=a(1+r)^{x}\\ y&=1,\!000(1+100\%)^{14}\\ y &= 1000(1+1)^{14}\\ y&=1,\!000\times2^{14}\\ y&=16,\!384,\!000 \end{align}

After two weeks, there were 16,384,000 viewers.