To determine which of the following options is equivalent to [tex]\( 8^{-5} \times 8^2 \)[/tex], we will use the properties of exponents. Specifically, we will use the rule that states:
[tex]\[ a^m \times a^n = a^{m+n} \][/tex]
In this problem, we have:
[tex]\[ 8^{-5} \times 8^2 \][/tex]
Here, [tex]\( a = 8 \)[/tex], [tex]\( m = -5 \)[/tex], and [tex]\( n = 2 \)[/tex]. We apply the exponent rule:
[tex]\[ 8^{-5} \times 8^2 = 8^{-5 + 2} \][/tex]
Now, add the exponents:
[tex]\[ -5 + 2 = -3 \][/tex]
So the expression simplifies to:
[tex]\[ 8^{-3} \][/tex]
Next, recall that a negative exponent indicates a reciprocal. That is:
[tex]\[ 8^{-3} = \frac{1}{8^3} \][/tex]
Therefore, the expression [tex]\( 8^{-5} \times 8^2 \)[/tex] is equivalent to [tex]\(\frac{1}{8^3}\)[/tex].
The correct answer is:
A) [tex]\(\frac{1}{8^3}\)[/tex]
