### Key Concepts

### 1.1 Real Numbers: Algebra Essentials

- Rational numbers may be written as fractions or terminating or repeating decimals. See Example 1 and Example 2.
- Determine whether a number is rational or irrational by writing it as a decimal. See Example 3.
- The rational numbers and irrational numbers make up the set of real numbers. See Example 4. A number can be classified as natural, whole, integer, rational, or irrational. See Example 5.
- The order of operations is used to evaluate expressions. See Example 6.
- The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See Example 7.
- Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See Example 8. They take on a numerical value when evaluated by replacing variables with constants. See Example 9, Example 10, and Example 12
- Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See Example 11 and Example 13.

### 1.2 Exponents and Scientific Notation

- Products of exponential expressions with the same base can be simplified by adding exponents. See Example 1.
- Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See Example 2.
- Powers of exponential expressions with the same base can be simplified by multiplying exponents. See Example 3.
- An expression with exponent zero is defined as 1. See Example 4.
- An expression with a negative exponent is defined as a reciprocal. See Example 5 and Example 6.
- The power of a product of factors is the same as the product of the powers of the same factors. See Example 7.
- The power of a quotient of factors is the same as the quotient of the powers of the same factors. See Example 8.
- The rules for exponential expressions can be combined to simplify more complicated expressions. See Example 9.
- Scientific notation uses powers of 10 to simplify very large or very small numbers. See Example 10 and Example 11.
- Scientific notation may be used to simplify calculations with very large or very small numbers. See Example 12 and Example 13.

### 1.3 Radicals and Rational Exponents

- The principal square root of a number $\phantom{\rule{0.8em}{0ex}}a\phantom{\rule{0.8em}{0ex}}$ is the nonnegative number that when multiplied by itself equals $\phantom{\rule{0.8em}{0ex}}a.\phantom{\rule{0.8em}{0ex}}$ See Example 1.
- If $\phantom{\rule{0.8em}{0ex}}a\phantom{\rule{0.8em}{0ex}}$ and $\phantom{\rule{0.8em}{0ex}}b\phantom{\rule{0.8em}{0ex}}$ are nonnegative, the square root of the product $\phantom{\rule{0.8em}{0ex}}ab\phantom{\rule{0.8em}{0ex}}$ is equal to the product of the square roots of $\phantom{\rule{0.8em}{0ex}}a\phantom{\rule{0.8em}{0ex}}$ and $\phantom{\rule{0.8em}{0ex}}b\phantom{\rule{0.8em}{0ex}}$ See Example 2 and Example 3.
- If $\phantom{\rule{0.8em}{0ex}}a\phantom{\rule{0.8em}{0ex}}$ and $\phantom{\rule{0.8em}{0ex}}b\phantom{\rule{0.8em}{0ex}}$ are nonnegative, the square root of the quotient $\phantom{\rule{0.8em}{0ex}}\frac{a}{b}\phantom{\rule{0.8em}{0ex}}$ is equal to the quotient of the square roots of $\phantom{\rule{0.8em}{0ex}}a\phantom{\rule{0.8em}{0ex}}$ and $\phantom{\rule{0.8em}{0ex}}b\phantom{\rule{0.8em}{0ex}}$ See Example 4 and Example 5.
- We can add and subtract radical expressions if they have the same radicand and the same index. See Example 6 and Example 7.
- Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See Example 8 and Example 9.
- The principal
*n*th root of $\phantom{\rule{0.8em}{0ex}}a\phantom{\rule{0.8em}{0ex}}$ is the number with the same sign as $\phantom{\rule{0.8em}{0ex}}a\phantom{\rule{0.8em}{0ex}}$ that when raised to the*n*th power equals $\phantom{\rule{0.8em}{0ex}}a.\phantom{\rule{0.8em}{0ex}}$ These roots have the same properties as square roots. See Example 10. - Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See Example 11 and Example 12.
- The properties of exponents apply to rational exponents. See Example 13.

### 1.4 Polynomials

- A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See Example 1.
- We can add and subtract polynomials by combining like terms. See Example 2 and Example 3.
- To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See Example 4.
- FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See Example 5.
- Perfect square trinomials and difference of squares are special products. See Example 6 and Example 7.
- Follow the same rules to work with polynomials containing several variables. See Example 8.

### 1.5 Factoring Polynomials

- The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See Example 1.
- Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See Example 2.
- Trinomials can be factored using a process called factoring by grouping. See Example 3.
- Perfect square trinomials and the difference of squares are special products and can be factored using equations. See Example 4 and Example 5.
- The sum of cubes and the difference of cubes can be factored using equations. See Example 6 and Example 7.
- Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See Example 8.

### 1.6 Rational Expressions

- Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See Example 1.
- We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See Example 2.
- To divide rational expressions, multiply by the reciprocal of the second expression. See Example 3.
- Adding or subtracting rational expressions requires finding a common denominator. See Example 4 and Example 5.
- Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See Example 6.