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A Fast Way to Solve Exponent Problems

Exponents are a component of a mathematical operation called exponentiation. Exponentiation involves two numbers including a base and an exponent. The process of exponentiation involves repeated multiplication of the base. The base number is multiplied by itself a number of times defined by the value of the exponent. For instance, the term 4^3 is equivalent to (4 * 4 * 4) = (16 * 4) = 64. Though it is easiest to use a scientific calculator to solve such problems, there are simplifications that can be performed on the expression to make it easier to solve by hand or calculator.

Instructions

Product Rule for Exponents
- 1
  Determine the smallest factor of the exponent. For example, for the power expression 4^25 the smallest factor that "goes into" the exponent is 5.
- 2
  Divide the exponent by its smallest factor to determine how many times the factor must be summed to equal the value of the exponent. For example, (25 / 5) = 5, so (5 + 5 + 5 + 5 + 5) = 25.
- 3
  Substitute the factor form of the exponent into the power expression. For example, 4^25 becomes 4^(5 + 5 + 5 + 5 + 5).
- 4
  Distribute the base number of the expression to each factor of the exponent and multiply them. For example, 4^(5 + 5 + 5 + 5 + 5) becomes (4^5 * 4^5 * 4^5 * 4^5 * 4^5). Since 4^5 = 1024 the expression becomes: (1024 * 1024 * 1024 * 1024 * 1024).
- 5
  Solve the expression. For example, using a calculator to solve (1024 * 1024 * 1024 * 1024 * 1024) finds the value to be ~1.126 * 10^15; the same value of 4^25.
Power Rule for Exponents
- 6
  Factorize the exponent into its prime factors. For example, for the expression 4^36, 36 has two prime factors: 2 and 3. And 36 is equivalent (2 * 2 * 3 * 3).
- 7
  Raise the base number of the expression to each prime factor, in succession. For example, 4^36 becomes 4^(2 * 2 * 3 * 3) becomes ((((4^2)^2)^3)^3).
- 8
  Solve the successive powers, beginning with the innermost expression. For example, ((((4^2)^2)^3)^3) = (((16)^2)^3)^3) = ((256)^3)^3) = (16777216^3) = (4.722 * 10^21). This is equivalent to 4^36.

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