Question

What index should the root form of the expression 1258⋅12−12have?(1 point)

Answer 1

To find the root form of the expression \( 1258 \cdot 12 - 12 \), we first simplify the expression:

1. Calculate \( 1258 \cdot 12 \):
\[
1258 \cdot 12 = 15096
\]

2. 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 \( 15084 \):

1. 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.