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Writing Quantities with a Common Base

We cannot multiply quantities if they do not have the same base. However, we can often use the "power of a power" rule to write two quantities as quantities with a common base. If the two bases are powers of a common base (for example, 4 and 32 are both powers of 2), rewrite each base as a power of the common base. Then uses the power of a power rule to combine the exponents of each base. Finally, use the multiplication rule and add the exponents.

Example: 95×274 =?

95 = (32)5 = 310; 274 = (33)4 = 312
95×274 = 310×312 = 310+12 = 322.

Summary of Properties

Addition: cxa + dxa = (c + d )xa
Subtraction: cxa - dxa = (c - d )xa
Multiplication: cxa×dxb = (cd )xa+b
Division: = ()xa-b
Distribution: (cd )a = cada
Power of a Power: c(xa)b = cxab