Question #70fcd

1 Answer
Jacob F.
Aug 2, 2014

If by 'distribute,' you mean #2*(2 + 4)^x ≟ (2*(2 + 4))^x#, then the answer is no, you cannot distribute the #2# in either case.

Remember that exponentiation is repeated multiplication. The first example, #2*(2+4)^3#, can be rewritten as

#2⋅(2+4)⋅(2+4)⋅(2+4)#

which is the same thing as

#2⋅6⋅6⋅6#

If instead of leaving the 2 outside, we distribute it within the parenthesis, then the following occurs:

#(2⋅(2+4))^3#

# = (2(2+4))⋅(2(2+4))⋅(2(2+4))#

# = 2(6)⋅2(6)⋅2(6)#

or, alternatively:

# = 2 * 2 * 2 * 6 * 6 * 6#

# = 2^3 * (2 + 4)^3#

which is clearly different from #2*(2 + 4)^3#.

You cannot distribute within something raised to a power because due to order of operations, you evaluate the parenthesis, then raise that to its exponent, and then multiply by the coefficient once. When you distribute like that, you're causing the coefficient to be multiplied many times, instead of just once. This produces something different than what you had originally.

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