Key Concepts

This is read $a$ to the $m^{\mathrm{th}}$ power.

• If $a$ is a real number and $m,n$ are counting numbers, then
  $$a^m·a^n=a^{m+n}$$
• To multiply with like bases, add the exponents.

• If $a$ is a real number and $m,n$ are counting numbers, then
  $${\left(a^m\right)}^n=a^{m\cdot n}$$

• If $a$ and $b$ are real numbers and $m$ is a whole number, then
  $${(ab)}^m=a^m{b}^m$$

• Distributive Property
• FOIL Method
• Vertical Method

• Distributive Property
• Vertical Method

• If $a,b,c$ are whole numbers where $b\ne 0,c\ne 0,$ then
  $$\frac{a}{b}=\frac{a·c}{b·c}\text{and}\frac{a·c}{b·c}=\frac{a}{b}$$

• If $a$ is a non-zero number, then $a^0=1.$
• Any nonzero number raised to the zero power is $1.$

• If $a$ is a real number, $a\ne 0,$ and $m,n$ are whole numbers, then
  $$\frac{a^m}{a^n}=a^{m-n},m>n\text{and}\frac{a^m}{a^n}=\frac{1}{a^{n-m}},n>m$$

• If $a$ and $b$ are real numbers, $b\ne 0,$ and $m$ is a counting number, then
  $${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$$
• To raise a fraction to a power, raise the numerator and denominator to that power.

• If $a,b$ are real numbers and $m,n$ are integers, then
  $$\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & a^m·a^n=a^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left(a^m\right)}^n=a^{m·n}\hfill \\ \mathbf{\text{Product to a Power Property}}\hfill & & & {(ab)}^m=a^m{b}^m\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & \frac{a^m}{a^n}=a^{m-n},a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill & & & a^0=1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & {\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m},b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill & & & a^{-n}=\frac{1}{a^n}\hfill \end{array}$$

1. Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Step 2. Count the number of decimal places, $n$, that the decimal point was moved.
3. Step 3.
   Write the number as a product with a power of 10.

   • If the original number is greater than 1, the power of 10 will be ${10}^n$.
   • If the original number is between 0 and 1, the power of 10 will be ${10}^{-n}$.
4. Step 4. Check.

1. Step 1. Determine the exponent, $n$, on the factor 10.
2. Step 2.
   Move the decimal $n$ places, adding zeros if needed.

   • If the exponent is positive, move the decimal point $n$ places to the right.
   • If the exponent is negative, move the decimal point $\left|n\right|$ places to the left.
3. Step 3. Check.

1. Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
2. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
3. Step 3. Bring down the common factors that all expressions share.
4. Step 4. Multiply the factors.

• If $a$, $b$, $c$ are real numbers, then
  $a(b+c)=ab+ac$ and $ab+ac=a(b+c)$

1. Step 1. Find the GCF of all the terms of the polynomial.
2. Step 2. Rewrite each term as a product using the GCF.
3. Step 3. Use the Distributive Property ‘in reverse’ to factor the expression.
4. Step 4. Check by multiplying the factors.