The Algebra of Exponents

Listed below are a number of problems that provide a summary of the basic properties of exponents. In each case, simplify the given expressions by combining identical factors, eliminating negative exponents, and actually computing the value of “small” numeric expressions (meaning the result is less than 1000).

  1. Rewrite each of the expressions below in either exponential or expanded notation, then compute the value of the expression. (Focus on what the exponent actually touches!)

a) / b)
c) / d)
e) / f)
  1. Use the “Product Rule” (multiplying like bases means add exponents) to simplify the expressions below. If you cannot use this rule to simplify something, write “Does Not Apply.”

a) / b)
c) / d)
  1. Use the “Quotient Rule” (dividing like bases means subtract exponents in the order top – bottom, and write your answer to this in the numerator before continuing to simplify) to simplify the expressions below. If you cannot use this rule, write “Does Not Apply.”

a) / b)
c) / d)
  1. Use the “Negative Exponent Rule” (eliminate the negative exponent by flipping the base from one side of the fraction bar to the other and changing the exponent’s sign) to simplify the expressions below.

a) / b)
c) / d)
  1. Use the “Power Rule” (raising exponential expressions to powers means multiply exponents) to simplify the expressions below. If you cannot use this rule to simplify something, write “Does Not Apply.”

a) / b)
c) / d)
  1. Use the “Distributing Rule for Exponents” (if bases are multiplied or divided in parentheses raised to a power, you use the Power Rule on each base inside parentheses) to simplify the expressions below. If you cannot use this, write “Does Not Apply.”

a) / b)
c) / d)
  1. Use the previous rules to simplify these expressions involving variables. You’ve finished when each base appears only once, all exponents are positive, and numbers to small powers are actually computed. (meaning the results are less than 1000)

a) / b)
c) / d)
e) / f)
g) / h)
i) / j)
k) / l)
m) / n)

Part II – Scientific Notation

  1. Recognizing common numbers in scientific notation. For each number below, rewrite it using either scientific notation or “newspaper notation” (e.g. 140 million, 2.3 trillion, etc.).

a) / b)75.3 million
c)2.43 thousandths / d)
  1. Converting between standard and scientific notations – always remember that small numbers (decimals less than one) in standard notation have negative exponents in scientific notation, while large numbers have positive exponents. When the number is in scientific notation, the exponent tells you how far to move the decimal point.

a)0.0008034 / b)
c) / d)4,006
  1. Multiplying numbers in scientific notation – multiply the number parts, add the powers of 10, and increase or decrease the powers of 10 to move the decimal into the correct position.

a) / b)
c) / d)
  1. Dividing numbers in scientific notation – divide the number parts, subtract the powers of 10, and increase or decrease the powers of 10 to move the decimal into the correct position.

a) / b)
c) / d)

Answers:

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