Intermediate Algebra/Exponents

Template:Algebra Page An exponential function is a shortcut for multiplication, just as multiplication is a shortcut for addition. For example:

$5+5+5+5+5=5\times 5$

$6\times 6=6^2$
$7\times 7\times 7=7^3$

When multiplying exponents of the same base, simply add the exponents. For example:

$x^2\times x^5=x^7$

When distributing an exponent, multiply any exponents. For example:

$(x^3{)}^6=x^{18}$

Scientific Notation

Scientific notation makes use of exponents. It is often used for very large or very small numbers. It's easier to write 1,574,000,000,000,000 as 1.574 * 10^15. To convert from regular notation to scientific notation, find the leftmost nonzero digit. Count how many places away it is from the ones digit. If the digit was on the right of the ones digit, the exponent will be negative. If it was the ones digit, the exponent will be zero. For everything to the left it will be positive. Then, move the decimal place of the original number so that exactly one nonzero digit is on the left. Write down this new number and * 10^(exponent). You're done!

Properties of Exponents

1. Product Rule
   $b^m\cdot b^n=b^{n+m}$
   
2. Quotient Rule
   $b^n÷b^m=b^{n-m}$
   
3. Zero - Exponent Rule
   $b^0=1,b\ne 0$
   $0^0$is undefined
   $(-4{)}^2=1$
   $-4^2=-1$
   
4. Power Rule
   $(b^m{)}^n=b^{m\cdot n}$
   Example: $(x^3{)}^2=x^{3\cdot 2}$
   
5. Negative Exponent Rule
   $b^{-n}=$$\frac{1}{b^n}$
   
6. Product - to - powers
   $(ab{)}^m=a^m\cdot b^m$
   $a^m\cdot b^m=(ab{)}^m$
   
7. Quotient - to - powers
   $(\frac{a}{b}{)}^n=\frac{a^n}{b^n}$

Note: Common mistakes made
$(a+b{)}^m\ne a^m+b^m$
$(a-b{)}^n\ne a^n-b^n$