To solve the question of rewriting $\sqrt[6]{12}$:

The goal is to simplify $\sqrt[6]{12}$ by expressing it in a different form through operations on the numbers in the expression, specifically 6 and 12.

The correct approach is to rewrite the root expression in terms of an equivalent exponent:

$\sqrt[6]{12} = 12^{\frac{1}{6}}$

Now let’s examine each option in the question to determine which transformation correctly simplifies this expression.

1. Option 1: Divide 6 by 3, resulting in $2^{12}$.

   • This would mean $12^{\frac{1}{6}} = 2^{12}$, which is not accurate.

2. Option 2: Divide 12 by 3, resulting in $6^{4}$.

   • This would give us $\sqrt[6]{12} = 6^4$, which is also incorrect.

3. Option 3: Subtract 3 from 6, resulting in $3^{12}$.

   • This doesn’t match the structure needed for $\sqrt[6]{12}$.

4. Option 4: Subtract 3 from 12, resulting in $6^9$.

   • This would give $6^9$, which is not equivalent to $\sqrt[6]{12}$.

After analyzing each option, none directly match the correct method to simplify or rewrite $\sqrt[6]{12}$. However, the exercise appears to intend for a specific approach involving properties of exponents or logarithmic simplification, but none here yield a true equivalent simplification for $\sqrt[6]{12}$.

Would you like a further breakdown of possible simplification strategies for radical expressions?

Related Questions:

1. How do you simplify higher roots like $\sqrt[6]{x}$ for other expressions?
2. Can exponents be manipulated in other ways to simplify radical expressions?
3. Why is dividing the numbers within a radical usually ineffective for simplification?
4. What is the relationship between roots and fractional exponents in simplification?
5. How would you express $\sqrt[6]{12}$ as a decimal approximation?

Tip:

Remember that simplifying radical expressions often involves finding factors or rewriting them with fractional exponents.