Let's evaluate the expression [tex]\( 64^{-3/2} \)[/tex] step-by-step.
First, recall that [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex]. This means we can rewrite [tex]\( 64^{-3/2} \)[/tex] as:
[tex]\[ 64^{-3/2} = \frac{1}{64^{3/2}} \][/tex]
Now, let's simplify [tex]\( 64^{3/2} \)[/tex].
1. We know that:
[tex]\[ 64^{3/2} = \left(64^{1/2}\right)^3 \][/tex]
2. Calculate the square root of 64:
[tex]\[ 64^{1/2} = \sqrt{64} = 8 \][/tex]
3. Raise the result to the power of 3:
[tex]\[ 8^3 = 8 \times 8 \times 8 = 512 \][/tex]
So, [tex]\( 64^{3/2} = 512 \)[/tex].
Putting this back into our fraction, we get:
[tex]\[ 64^{-3/2} = \frac{1}{512} \][/tex]
Thus, the value of [tex]\( 64^{-3/2} \)[/tex] is:
[tex]\[ \boxed{\frac{1}{512}} \][/tex]
So, the correct option is (d) [tex]\( \frac{1}{512} \)[/tex].
