To simplify the expression [tex]\( d^3 \cdot d^5 \)[/tex], we will use the laws of exponents. Specifically, we use the rule that states when you multiply two exponential expressions with the same base, you add the exponents.
Here's the detailed step-by-step solution:
1. Start with the given expression:
[tex]\[
d^3 \cdot d^5
\][/tex]
2. Identify the base and the exponents. The base here is [tex]\( d \)[/tex], and the exponents are 3 and 5.
3. Apply the rule for multiplying exponents with the same base:
[tex]\[
d^a \cdot d^b = d^{a+b}
\][/tex]
In this case, [tex]\( a = 3 \)[/tex] and [tex]\( b = 5 \)[/tex].
4. Add the exponents together:
[tex]\[
3 + 5 = 8
\][/tex]
5. Consequently, the simplified expression is:
[tex]\[
d^8
\][/tex]
Therefore, the correct simplification of the expression [tex]\( d^3 \cdot d^5 \)[/tex] is [tex]\( d^8 \)[/tex].
Thus, the correct answer is:
[tex]\[
d^8
\][/tex]
