Exponents
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Contents
 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:
$\begin{align} a^{b}\div a^{c}&= {a^{bc}}\\ 5^{3} \div 5^{2} &=5^{32}\\ 5^{3} \div 5^{2} &=5^{1}\\ 125\div 25&=5^{1}\\ 5^{1}&=5 \end{align}$
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: $y=a(1r)^{x}$
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.
All Videos in this Topic
Videos in this Topic
Exponents (7 Videos)
All Worksheets in this Topic
Worksheets in this Topic
Exponents (7 Worksheets)

Zero and Negative Exponents
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Exponents and Multiplication – Product of Powers Property
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Exponents and Division – the Quotient of Powers Property
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Powers of Products and Quotients
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Reading and Writing Scientific Notation
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Operations with Numbers in Scientific Notation
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Exponential Growth and Decay
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