### Key Concepts

## 10.2 Use Multiplication Properties of Exponents

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

**Product Property of Exponents**- If $a$ is a real number and $m,n$ are counting numbers, then
$${a}^{m}\xb7{a}^{n}={a}^{m+n}$$
- To multiply with like bases, add the exponents.

- If $a$ is a real number and $m,n$ are counting numbers, then
**Power Property for Exponents**- If $a$ is a real number and $m,n$ are counting numbers, then $${\left({a}^{m}\right)}^{n}={a}^{m\cdot n}$$

- If $a$ is a real number and $m,n$ are counting numbers, then
**Product to a Power Property for Exponents**- If $a$ and $b$ are real numbers and $m$ is a whole number, then$${(ab)}^{m}={a}^{m}{b}^{m}$$

- If $a$ and $b$ are real numbers and $m$ is a whole number, then

## 10.3 Multiply Polynomials

**Use the FOIL method for multiplying two binomials.**Step 1. Multiply the **First**terms.Step 2. Multiply the **Outer**terms.Step 3. Multiply the **Inner**terms.Step 4. Multiply the **Last**terms.Step 5. Combine like terms, when possible. **Multiplying Two Binomials:**To multiply binomials, use the:- Distributive Property
- FOIL Method
- Vertical Method

**Multiplying a Trinomial by a Binomial:**To multiply a trinomial by a binomial, use the:- Distributive Property
- Vertical Method

## 10.4 Divide Monomials

**Equivalent Fractions Property**- If $a,\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}c$ are whole numbers where $b\ne 0,\phantom{\rule{0.2em}{0ex}}c\ne 0,$ then
$$\frac{a}{b}=\frac{a\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}c}{b\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}c}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{a\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}c}{b\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}c}=\frac{a}{b}$$

- If $a,\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}c$ are whole numbers where $b\ne 0,\phantom{\rule{0.2em}{0ex}}c\ne 0,$ then
**Zero Exponent**- If $a$ is a non-zero number, then ${a}^{0}=1.$
- Any nonzero number raised to the zero power is $1.$

**Quotient Property for Exponents**- If $a$ is a real number, $a\ne 0,$ and $m,\phantom{\rule{0.2em}{0ex}}n$ are whole numbers, then $$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{2em}{0ex}}m>n\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\phantom{\rule{2em}{0ex}}n>m$$

- If $a$ is a real number, $a\ne 0,$ and $m,\phantom{\rule{0.2em}{0ex}}n$ are whole numbers, then
**Quotient to a Power Property for Exponents**- If $a$ and $b$ are real numbers, $b\ne 0,$ and $m$ is a counting number, then $${\left(\frac{a}{b}\right)}^{m}=\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{b}^{m}}$$
- To raise a fraction to a power, raise the numerator and denominator to that power.

- If $a$ and $b$ are real numbers, $b\ne 0,$ and $m$ is a counting number, then

## 10.5 Integer Exponents and Scientific Notation

**Summary of Exponent Properties**- If $a,b$ are real numbers and $m,n$ are integers, then
$$\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & {a}^{m}\xb7{a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left({a}^{m}\right)}^{n}={a}^{m\xb7n}\hfill \\ \mathbf{\text{Product to a Power Property}}\hfill & & & {(ab)}^{m}={a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{0.2em}{0ex}}a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill & & & {a}^{0}=1,\phantom{\rule{0.2em}{0ex}}a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & {\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},\phantom{\rule{0.2em}{0ex}}b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill & & & {a}^{-n}=\frac{1}{{a}^{n}}\hfill \end{array}$$

- If $a,b$ are real numbers and $m,n$ are integers, then
**Convert from Decimal Notation to Scientific Notation:**To convert a decimal to scientific notation:- Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
- Step 2. Count the number of decimal places, $n$, that the decimal point was moved.
- 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 ${10}^{-n}$.

- Step 4. Check.

**Convert Scientific Notation to Decimal Form:**To convert scientific notation to decimal form:- Step 1. Determine the exponent, $n$, on the factor 10.
- 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.

- Step 3. Check.

## 10.6 Introduction to Factoring Polynomials

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

**Distributive Property**- If $a$, $b$, $c$ are real numbers, then

$a(b+c)=ab+ac$ and $ab+ac=a(b+c)$

- If $a$, $b$, $c$ are real numbers, then
**Factor the greatest common factor from a polynomial.**- Step 1. Find the GCF of all the terms of the polynomial.
- Step 2. Rewrite each term as a product using the GCF.
- Step 3. Use the Distributive Property ‘in reverse’ to factor the expression.
- Step 4. Check by multiplying the factors.