Using the properties of exponents to generate an equivalent expression, which of the following would be the rational exponent on an expression equivalent to 3216? (1 point)

Question

-1/3
-3
3
1/3

Answer 1

To rewrite \( 3216 \) using properties of exponents, we need to express it in a form that involves the base \( 3 \).

First, we can rewrite \( 3216 \) in terms of its prime factors:

\[ 3216 = 3 \times 1072 \] Now we can further factor \( 1072 \):

\[ 1072 = 16 \times 67 = 2^4 \times 67 \]

Thus, we can write \( 3216 \) as:

\[ 3216 = 3 \times 2^4 \times 67 \]

Now, if we consider the expression \( 3^{216} \) instead, we notice that:

When expressing any base with a power of \( n \), it is equal to \( x^{mn} \) where \( m \) is the exponent and \( n \) is related to the base.

Since we only need to evaluate if the expression \( 3^{216} \) can be represented with rational exponents, we note that:

\[ 3^{216} = (3^1)^{216} = 3^{1 \cdot 216} = 3^{216} \]

Now, since none of the answer choices appear to be simplifications or representations of \( 3^{216} \), we need to analyze the choices.

Considering that the form \( 3^{k} \) should produce those options:

• The value of the rational exponent equivalent to \( 3216 \) cannot be directly evident.

If we look for a rational exponent scenario involving cube roots or square roots, we need to find the equivalent expressions resulting in the options presented.

The only option relevant to working with or comparing with \( 3^{216} \) terms is relating \( k \) with fractional exponents and the appropriate value.

However, in this case, there is no direct mathematical transformation provided that shows a consistent energy or conversion related back to \( 3216 \) until clarified.

Given typical characteristic, if we are simply looking for basic rational relationships attributed to generated numbers, we also consider the various options.

The best assumption from potential continuity, rational conversion, or obstructed values presents that:

The answer would typically correspond closest to \( 3 \) or \( 1/3 \) in fractional mapping, allowing some flexibility.

Hence, the correct answer equivalent to the expression \( 3216 \) using rational exponents given lacks clarity directly and indicates asking clarification on \( 3^{k} \), leaving us likely with None directly beyond \( k \).

Please clarify application, approximation, or consider adjustments as variables quantify further!