Question

What index should the root form of the expression 1258⋅12−12
have?

Answer 1

To solve the expression \( 1258 \cdot 12 - 12 \) and determine what index the root form should have, we first simplify it:

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

2. Now, subtract \( 12 \):
\[
15096 - 12 = 15084
\]

Next, we consider the square root form of the expression. We want to express \( 15084 \) in a root format. Since you're asking about the "index" of the root form, the most common root forms are square roots (index of 2) or cube roots (index of 3), but generally, the index can be any positive integer.

To determine if we can simplify \( 15084 \):
- We factor \( 15084 \) to find out if it has a perfect square component.

The prime factorization of \( 15084 \) is:
\[
15084 = 2^2 \cdot 3 \cdot 1257
\]
Continuing with \( 1257 \):
- We find \( 1257 = 3 \cdot 419 \) - and checking \( 419 \):
- \( 419 \) is prime.

So, the complete factorization:
\[
15084 = 2^2 \cdot 3^2 \cdot 419
\]

To maximize the root, we can take the square root:
\[
\sqrt{15084} = 2 \cdot 3 \cdot \sqrt{419} = 6 \sqrt{419}
\]

Therefore, for the root form of the expression, the index we are primarily considering is **2**, as we typically refer to square roots first unless specified otherwise.

Thus, the root form of the expression \( 1258 \cdot 12 - 12 \) has an index of \( 2 \).