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Define base and exponent, and express numbers as products of prime factors in exponential form.
Introduce base, power, and notation.
Imagine folding a large sheet of paper over and over. If you fold it 10 times, the thickness increases by times.
Instead of writing this long multiplication string, mathematics uses a powerful shorthand called exponential notation.
Formal definition of exponential notation.
Repeated multiplication of the same number.
In this definition, is a counting number:
The base is multiplied by itself times.
Converting a large number to exponential form.
Problem. Express the number 32400 as a product of its prime factors and represent the prime factors in their exponential form.
Faded example for prime factorization.
Let's express 648 as a product of powers of its prime factors in exponential form. First, we find the prime factorization of 648 by dividing by its smallest prime factor, 2.
Dividing by 2 repeatedly gives: 648 = 2 × 324 = 2 × 2 × 162 = 2 × 2 × 2 × 81. Next, we factor 81 by dividing by its smallest prime factor, 3. Dividing by 3 repeatedly gives: 81 = 3 × 27 = 3 × 3 × 9 = 3 × 3 × 3 × 3.
Combining these steps, the full prime factorization is 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3. To write this in exponential form, we count the number of times each base is multiplied by itself.
Because 2 is multiplied 3 times and 3 is multiplied 4 times, the final exponential form is 648 = 2^ × 3^.
Testing exponents with negative bases.
What is the value of ?