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 \( \frac{3}{2} \).
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