How do you evaluate #64^ { 4/ 3} \cdot 64^ { - 3/ 2}#?

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Answer 1 · 2017-01-28T17:47:02

See the entire evaluation process below:

First, transform the exponents to have common denominators by multiplying by the appropriate form of #1#:

#64^(4/3) * 64^(-3/2) -> 64^(4/3 * 2/2) * 64^(-3/2 * 3/3) ->#

#64^(8/6) * 64^(-9/6)#

Next, we can combine these terms using this rule of exponents:

#x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) +color(blue)(b))#

#64^(8/6 + -9/6) -> 64^(-1/6)#

We can now use this rule of exponents to further modify the expression:

#x^color(red)(a) = 1/x^color(red)(-a)#

#1/64^(- -1/6) -> 1/64^(1/6)#

Now, we can use this final rule of exponents to complete the evaluation of this expression:

#x^(1/color(red)(n)) = root(color(red)(n))(x)#

#1/root(color(red)(6))(64) ->#

#1/2#