Sure, let's solve the problem step-by-step.
First, we need to interpret the given expression [tex]\( 10 \times 8^{-2} \)[/tex].
1. Understanding Negative Exponent: The expression [tex]\( 8^{-2} \)[/tex] means [tex]\( \frac{1}{8^2} \)[/tex].
2. Calculating the Positive Exponent: Now, let's find [tex]\( 8^2 \)[/tex]:
[tex]\[ 8^2 = 8 \times 8 = 64 \][/tex]
3. Applying the Negative Exponent: Therefore,
[tex]\[ 8^{-2} = \frac{1}{8^2} = \frac{1}{64} \][/tex]
4. Multiplying by 10: Now multiply this result by 10:
[tex]\[ 10 \times 8^{-2} = 10 \times \frac{1}{64} = \frac{10}{64} \][/tex]
5. Simplifying the Fraction: Finally, we need to simplify the fraction [tex]\( \frac{10}{64} \)[/tex]. To do this, we find the greatest common divisor (GCD) of 10 and 64. We know that the GCD is 2.
Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{10 \div 2}{64 \div 2} = \frac{5}{32}
\][/tex]
Therefore, [tex]\( 10 \times 8^{-2} \)[/tex] as a fraction in its simplest form is [tex]\( \frac{5}{32} \)[/tex].
