To find the root form of the expression \( 1258 \cdot 12 - 12 \), we first simplify the expression:
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Calculate \( 1258 \cdot 12 \): \[ 1258 \cdot 12 = 15096 \]
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Now subtract \( 12 \): \[ 15096 - 12 = 15084 \]
Next, we can express \( 15084 \) in root form:
If you're looking for the index (or exponent) for the root form, we need to find the prime factorization of \( 15084 \):
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Start dividing by small prime numbers:
15084 is even, so divide by 2: \[ 15084 \div 2 = 7542 \]
7542 is also even: \[ 7542 \div 2 = 3771 \]
Now, check for divisibility by 3 (sum of digits \( 3 + 7 + 7 + 1 = 18 \) which is divisible by 3): \[ 3771 \div 3 = 1257 \] Check 1257 (sum of digits \( 1 + 2 + 5 + 7 = 15 \)): \[ 1257 \div 3 = 419 \] Next, check \( 419 \) (which is a prime number).
Collecting our factors gives us: \[ 15084 = 2^2 \cdot 3^2 \cdot 419^1 \]
Now, if we express this in root form, we could write it as: \[ \sqrt[n]{15084} = 2^{2/n} \cdot 3^{2/n} \cdot 419^{1/n} \]
If you are asking for the 'index' of the root form, it would depend on how you want to represent it (typically denoted as \( n \)). The simplest form is \( n = 2 \) for the square root.
Thus, to summarize, if you require the expression in root form with \( n = 2 \): \[ \sqrt{15084} = 2 \cdot 3 \cdot \sqrt{419} = 6\sqrt{419} \]
If you need a specific numerical value for \( n \), please clarify. However, for the common interpretation, the root form would involve an index (which is the value of \( n \), commonly set to 2) when taking a square root.
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