To simplify the problem \( 12^{-5} \cdot 12^{7} \), we can follow these steps:
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Understand Exponents: First, we need to remember that when we multiply numbers with the same base (in this case, 12), we can add their exponents together. Exponents are the little numbers that tell us how many times to multiply the base by itself.
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Add the Exponents: In our problem, the exponents are \(-5\) and \(7\). So, we will add those two numbers together: \(-5 + 7\).
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Calculate the Sum: When we add \(-5\) and \(7\), we get \(2\).
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Write the Result: Now, we can write our expression using the new exponent we calculated. Instead of \( 12^{-5} \cdot 12^{7} \), we can write it as \( 12^{2} \).
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Final Answer: So, the simplified answer is \( 12^{2} \), which means \( 12 \times 12 \).
In conclusion, the problem \( 12^{-5} \cdot 12^{7} \) simplifies to \( 12^{2} \), and if we wanted to, we could even find the value of \( 12^{2} \), which is \( 144 \).
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