Question
Using the Product of Powers Property, simplify: \( a^{3} \cdot a^{5} \)
Ask by Warren Cox. in Turkey
Jan 29,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( a^{3} \cdot a^{5} = a^{8} \)
Solution
To simplify \( a^{3} \cdot a^{5} \) using the **Product of Powers Property**, follow these steps:
### **Product of Powers Property:**
When you multiply two expressions that have the same base, you **add** their exponents.
\[
a^{m} \cdot a^{n} = a^{m + n}
\]
### **Applying the Property:**
Given:
\[
a^{3} \cdot a^{5}
\]
Using the Product of Powers Property:
\[
a^{3} \cdot a^{5} = a^{3 + 5} = a^{8}
\]
### **Final Answer:**
\[
a^{3} \cdot a^{5} = a^{8}
\]
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Simplify this solution Beyond the Answer
When you multiply powers with the same base, you simply add their exponents! So, for \( a^{3} \cdot a^{5} \), you combine the exponents like this: \( 3 + 5 = 8 \). Therefore, the simplified expression is \( a^{8} \). It's a neat little shortcut, right? This property applies no matter how big the exponents are, so whether you're looking at \( a^{100} \cdot a^{200} \) or something more manageable, you can always keep it simple by just adding the exponents!
