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Understanding Monomials and Exponents in Mathematics

Learn how to multiply, divide, and simplify monomials with exponents. Follow step-by-step procedures and rules for efficient calculations. Examples and practice problems included.

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Understanding Monomials and Exponents in Mathematics

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  1. Multiplying Monomials Multiplication can always be performed between two factors. Exponents will change when two like variable factors are being multiplied. To multiply two like variable factors: Product Rule of Exponents Procedure: To Multiply monomials Step 1. Multiply numerical coefficients Step 2. Multiply like variables one at a time, in alphabetical order. Solution: Answer: Your Turn Problem #1

  2. Solution: Answer: Your Turn Problem #2 An exponent written immediately following a parenthesis indicates the number of times the term within the parentheses is being multiplied by itself. Power to a Power Rule of Exponents Examples: Recall: If no exponent is shown, the understood exponent is 1.

  3. Combining the two rules for exponents. Examples: Procedure: To simplify expressions with an exponent outside and following parenthesis: Step 1: Multiply all exponents inside parenthesis by the exponent outside parenthesis. Step 2: Write the product of Step 1 as the exponent of each variable in the answer. Step 3: Multiply out the numerical coefficient. Solution: Answer: Your Turn Problem #3

  4. Note: When a negative factor is inside the parentheses, and the exponent on the outside is: even: the result is positive odd: the result is negative Examples: Solution: Answer: Your Turn Problem #4

  5. Dividing Monomials Recall from arithmetic, a fraction that is equal to 1 contains a numerator that is equal to its denominator. For example: Before we give some formal rules for dividing monomials, let’s perform the following with our understanding of exponents. Note: When the denominator equals 1, it does not need to be written. Rewrite both numerator and denominator without exponents. Divide out pairs of identical factors, one from the numerator and one from the denominator. Each factor of the pair is lined out and converted to an understood “1”. When all factors in the numerator divide out, the numerator equals “1” which must be written.

  6. Quotient Rule of Exponents If b is any nonzero real number, and m and n are positive integers, then Examples: In other words, find the difference between the exponents. Keep the variable where exponent is larger. If the exponent in the numerator is larger, keep the variable in the numerator. If the exponent in the denominator is larger, keep the variable in the denominator. In example c, anything divided by itself equals 1.

  7. Procedure: To divide two monomials Step 1. Reduce the numerical coefficients. Step 2. Taking each variable type separately, divide out as in the previous slides. 8 Answer: Your Turn Problem #5 Step 1. Reduce coefficients Step 2. Subtract exponents for like variables

  8. 12 Your Turn Problem #6 Therefore the answer is: • Simplify coefficients By definition any real number with an exponent of 0 is equal to 1. • Subtract the exponents for each variable

  9. Your Turn Problem #7 Simplify: Answer: The End. B.R. 12-15-07 Example 7. Simplify: • Simplify coefficients. • Subtract the exponents for each variable Therefore the answer is:

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