Using the properties of exponents to generate an equivalent expression, which of the following would be the rational exponent on an expression equivalent to 3^√216 ?

1 answer

To express \( 3^{\sqrt{216}} \) using rational exponents, we first need to simplify \( \sqrt{216} \).

We can factor \( 216 \) as follows:

\[ 216 = 2^3 \times 3^3 \]

Now, we can find the square root:

\[ \sqrt{216} = \sqrt{2^3 \times 3^3} = \sqrt{2^3} \times \sqrt{3^3} = 2^{3/2} \times 3^{3/2} \]

However, we will realize it is easier to calculate just the square root itself.

Calculating \( \sqrt{216} \):

\[ \sqrt{216} = \sqrt{36 \times 6} = \sqrt{36} \times \sqrt{6} = 6\sqrt{6} \]

Now we can rewrite the expression \( 3^{\sqrt{216}} \):

\[ 3^{\sqrt{216}} = 3^{6\sqrt{6}} \]

Now, converting \( 6\sqrt{6} \) to a rational exponent:

Recall, \( \sqrt{6} \) can be expressed as \( 6^{1/2} \):

So,

\[ 6 \sqrt{6} = 6^{1} \cdot 6^{1/2} = 6^{1 + 1/2} = 6^{3/2} \]

Thus, we have:

\[ 3^{\sqrt{216}} = 3^{6\sqrt{6}} = 3^{6^{3/2}} = 3^{(6^{3/2})} = 3^{6^{3/2}} \]

In this context, the rational exponent expressed for the square root can be written as:

\[ 3^{(216^{1/2})} \]

Thus, the rational exponent on the expression equivalent to \( 3^{\sqrt{216}} \) is

\[ \frac{3}{2} \]

Final expression: The rational exponent is \( \frac{3}{2} \).