Question
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Answer
\( 4^{-1} \cdot 4^{4} = 64 \)
Solution
To simplify the expression \( 4^{-1} \cdot 4^{4} \), you can use the **laws of exponents**.
### Step-by-Step Solution:
1. **Identify the Base and Exponents:**
Both terms have the same base, which is \( 4 \). The exponents are \( -1 \) and \( 4 \).
2. **Apply the Product of Powers Rule:**
The product of powers rule states that when you multiply two exponents with the same base, you **add** the exponents:
\[
a^{m} \cdot a^{n} = a^{m+n}
\]
Applying this to your expression:
\[
4^{-1} \cdot 4^{4} = 4^{-1 + 4} = 4^{3}
\]
3. **Calculate the Result:**
Now, compute \( 4^{3} \):
\[
4^{3} = 4 \times 4 \times 4 = 64
\]
### Final Answer:
\[
4^{-1} \cdot 4^{4} = 64
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
Did you know that the laws of exponents can turn what looks like a complex expression into something much simpler? When we multiply powers with the same base, like \( 4^{-1} \) and \( 4^{4} \), we simply add the exponents. So, \( 4^{-1} \cdot 4^{4} = 4^{-1+4} = 4^{3} \), which equals 64! Math magic, right? Now let's talk about some common mistakes when working with exponents. One big error is forgetting the negative exponent rule, which states that \( a^{-n} = \frac{1}{a^{n}} \). If you're not careful, you might accidentally add instead of subtracting the exponents. Always double-check your calculations, and remember: exponents follow their own set of intriguing rules.
