# Exponents

### Exponential Notation

Exponential notation can be used to represent repeated multiplication.
Exponential notation is used to show repeated multiplication of the same number. The exponent indicates how many times a number is multiplied by itself, shown by a raised number, such as:
\begin{aligned}4^3&=4\cdot4\cdot4\\&=64\end{aligned}
In the expression $4^3$, the 4 is the base, and the 3 is the exponent. The expression $4^3$, or the value 64, is the power. The expression is read "four to the third power." The value 64 is called "the third power of four."

There are some instances where exponents are either zero or a negative number. They follow a special set of rules that can be used to evaluate expressions.

### Zero and Negative Exponents

Rule Example
A nonzero number raised to the zero power is equal to 1.
$5^{0}= 1$
A nonzero number raised to a negative power is the reciprocal of that number raised to the opposite of the power.
\begin{aligned}5^{-2}&= \frac{1}{5^{2}}\\&= \frac{1}{25}\end{aligned}
Zero raised to any positive power is zero.
$0^5 = 0$
Zero to the zero power is generally defined as 1.

In certain contexts, however, $0^0$ is referred to as an indeterminate form. So, a calculator may show it as undefined.
$0^0=1$
Zero raised to a negative power is undefined.
$0^{-3}=\text{Undefined}$

### Scientific Notation

Scientific notation uses exponents to represent very large or very small numbers.
Scientific notation is useful in writing very small numbers, such as the mass of an electron at about 0.0000000000000000000000000000009109 kilograms, and very large numbers, such as the number of molecules in one mole at about 602,000,000,000,000,000,000,000 molecules in a mole. To calculate with such large numbers, rewrite them using scientific notation. Scientific notation is used to express a number as the product of two factors. The first is a number greater than or equal to 1 and less than 10. The second number is a power of 10. The exponent in the power of 10 indicates the number of places the decimal point moves when changing from standard form to scientific notation. So, the scientific notation of the mass of an electron is written as:
$0.0000000000000000000000000000009109=9.109\times 10^{-31} \; \text{kg per electron}$
The scientific notation for the number of molecules in one mole can be written as:
$602,000,000,000,000,000,000,000=6.02\times 10^{23} \; \text{molecules per mole}$

### Rules of Exponents

Rules of exponents can be used to simplify expressions containing exponents.

To simplify expressions containing exponents, it may be helpful to look at some expressions written in expanded form.

When an exponential expression is expanded and multiplied, exponents are added:
\begin{aligned}3^{{\color{#c42126}{5}}} \cdot3^{{\color{#c42126}{2}}}&=(3\cdot3\cdot3\cdot3\cdot3)(3\cdot3)\\&=3\cdot3\cdot3\cdot3\cdot3\cdot3\cdot3\\&=3^{{\color{#c42126}{7}}}\end{aligned}
When an exponential expression is expanded and divided, exponents are subtracted:
\begin{aligned}\frac{3^{\color{#c42126} 5}}{3^{\color{#c42126}2}}&=\frac{\cancel{3}\cdot\cancel{3}\cdot3\cdot3\cdot3}{\cancel{3}\cdot\cancel{3}}\\&=3\cdot3\cdot3\\&=3^{\color{#c42126}3}\end{aligned}
When an exponential expression is raised to a certain power, exponents are multiplied:
\begin{aligned}(3^{\color{#c42126} 5})^{\color{#c42126}2}&=(3\cdot3\cdot3\cdot3\cdot3)(3\cdot3\cdot3\cdot3\cdot3)\\&=3^{{\color{#c42126}10}}\end{aligned}
The process of expanding can be generalized to understand all of the rules of exponents.

### Properties of Exponents

Property Example Description
Product of powers property
$x^{m} \cdot x^{n} = x^{m+n}$
To multiply two powers with the same base, add the exponents.
Power of a product property
$(xy)^{m} = x^{m} \cdot y^{m}$
To raise a product to a power, raise both factors to the power.
Quotient of powers property
$\frac {x^{m}}{x^{n} } = x^{m-n}$
To divide two powers with the same base, subtract the exponents.
Power of a quotient property
$\left({\frac{x}{y}}\right)^{m}=\frac{x^m}{y^m}$
To raise a quotient to a power, raise both expressions to the power.
Power of a power property
$\left(x^{m}\right)^{n} = x^{mn}$
To raise a power to a power, multiply the exponents.

Step-By-Step Example
Simplifying an Exponential Expression
Simplify the expression:
$\frac{(x^5y^3)^2(xy^2)}{x^3y^2}$
Step 1

Simplify the power $(x^5y^3)^2$. Multiply the exponents.

Use the power of a product property to write $(x^5y^3)^2$ as $(x^5)^2(y^3)^2$.
$\begin{gathered}\frac{({\color{#c42126}{x^5}}{\color{#0047af}{y^3}})^\mathbf{2}(xy^{2})}{x^{3}y^{2}}\\\frac{{\color{#c42126}{(x^5)^{2}}}{\color{#0047af}{(y^3)^{2}}}(xy^{2})}{x^{3}y^{2}}\end{gathered}$
Step 2
Use the power of a power property to write $(x^5)^2$ as $x^{10}$ and $(y^3)^2$ as $y^6$.
$\begin{gathered}\frac{{\color{#c42126}{(x^5)^{2}}}{\color{#0047af}{(y^3)^{2}}}(xy^{2})}{x^{3}y^{2}}\\\frac{({\color{#c42126}{x^{10}}}{\color{#0047af}{y^6}})(xy^{2})}{x^{3}y^{2}}\end{gathered}$
Step 3
Multiply the powers in the numerator. Group the factors with a base of $x$ and the factors with a base of $y$ together.
\begin{aligned}\frac{({\color{#c42126}{x^{10}}}{\color{#0047af}{y^6}})({\color{#c42126}{x}}{\color{#0047af}{y^2}})}{x^{3}y^{2}}\\\frac{{\color{#c42126}{(x^{10}x)}}{\color{#0047af}{(y^{6}y^{2})}}}{x^{3}y^{2}}\end{aligned}
Step 4
Add the exponents. Use the product of powers property to write $x^{10}x$ as $x^{10+1}$ and $y^6y^2$ as $y^{6+2}$.
$\begin{gathered}\frac{{\color{#c42126}{(x^{10}x)}}{\color{#0047af}{(y^{6}y^{2})}}}{x^{3}y^{2}}\\\frac{{\color{#c42126}{(x^{10+1}})}{{\color{#0047af}{(y^{6+2}})}}}{x^{3}y^{2}}\end{gathered}$
Step 5
Simplify the expression.
$\begin{gathered}\frac{{\color{#c42126}{(x^{10+1}})}{{\color{#0047af}{(y^{6+2}})}}}{x^3y^2}\\\frac{{\color{#c42126}{(x^{11}})}{\color{#0047af}{(y^{8})}}}{x^3y^2}\end{gathered}$
Step 6
Use the quotient of powers property to write $\frac{x^{11}}{x^3}$ as $x^{11-3}$ and $\frac{y^{8}}{y^2}$ as $y^{8-2}$.
\begin{aligned}\frac{{\color{#c42126}{(x^{11})}}{\color{#0047af}{(y^{8})}}}{{\color{#c42126}{x^3}}{\color{#0047af}{y^2}}}\\{\color{#c42126}{x^{11-3}}}{\color{#0047af}{y^{8-2}}}\end{aligned}
Solution
Simplify the expression.
$\begin{gathered}{\color{#c42126}{x^{11-3}}}{\color{#0047af}{y^{8-2}}}\\{\color{#c42126}{x^8}}{\color{#0047af}{y^6}}\end{gathered}$
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