7th Grade Chapter 4

Chapter 4 - Factors, Fractions and Exponents

The product of two integers is an integer, and both integers are factors of the product. Moreover, both integers divide the product, and the product is said to be divisible by each integer.

An exponent is a number that is placed to the upper right of another number or expression called the base. For example, 2^5 is a power with an exponent 5 and a base 2. Sometimes this is read “two to the power of 5” or “two to the 5th power”. If an exponent has as its base an expression involving addition the whole expression must be written inside parentheses. Otherwise, the exponent has only the number next to it as its base.

Every prime number has exactly two factors and every composite number has at least three factors. The numbers 2 and 3 have exactly two factors and are prime. The number 4, for example, has factors of 1,2, and 4 and is composite.

When you multiply or divide both numerator and denominator of a fraction by the same nonzero number, you get a fraction that is equivalent to the original fraction. Equivalent fraction represent equal values. They name the same point on a number line. You can simplify a fraction by dividing the numerator and denominator by the GCF.

Sometimes a complicated problem can be solved by breaking it into simpler parts, then organizing the problem data into carefully constructed lists or diagrams and looking for patterns, If you do not draw diagrams neatly or construct lists systematically, it is easy to miss one of the possibilities, or to include a possibility more than once. Many of the problems in this lesson are part of a branch of mathematics called discrete math.

Rational numbers are defined as a ratio of two integers, where the denominator does not equal zero. For every rational number, there are an unlimited number of equivalent fractions. Using equivalent fractions, students can compare and order rational numbers.

To discover the rules of exponents write out all the factors of 5^3 x 5^2 and (5^3)^2. You find that to multiply powers with a common base, you add exponents. To raise a power to a power, you multiply exponents.

Exponent rules are given in terms of positive and integer exponents. Negative and zero exponents are defined to maintain consistency with these rules. Students apply these exponent rules in order to correctly multiply, divide, and simplify rational numbers.

Scientific notation provides a short form for very large and very small numbers. Scientific notation often uses the multiplication symbol x to avoid confusion with the decimal point in the the first factor. The exponent with 10 shows the number of places you move the decimal point. A positive exponent indicates that the number is greater than or equal to 1. A negative exponent indicates that the number is less than 1.