Exponents

Content

• Introduction
• Properties of Exponents
• Zero Exponent Property
• Negative Exponent Property
• Product of Powers Property
• Quotient of Powers Property
• Power of a Product Property
• Power of a Quotient Property
• Power of a Power Property
• Scientific Notation
• Exponential Growth and Decay

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.