To simplify the expression \( \sqrt{7} \times \sqrt[6]{7} \) using rational exponents, we start by expressing each root in terms of exponents.
- The square root \( \sqrt{7} \) can be written as \( 7^{1/2} \).
- The sixth root \( \sqrt[6]{7} \) can be written as \( 7^{1/6} \).
Now we can rewrite the expression:
\[ \sqrt{7} \times \sqrt[6]{7} = 7^{1/2} \times 7^{1/6} \]
Next, we apply the property of exponents that states \( a^m \times a^n = a^{m+n} \):
\[ 7^{1/2} \times 7^{1/6} = 7^{1/2 + 1/6} \]
To add \( \frac{1}{2} \) and \( \frac{1}{6} \), we need a common denominator. The least common multiple of 2 and 6 is 6. We convert \( \frac{1}{2} \):
\[ \frac{1}{2} = \frac{3}{6} \]
Now we can add:
\[ \frac{3}{6} + \frac{1}{6} = \frac{4}{6} \]
Simplifying \( \frac{4}{6} \):
\[ \frac{4}{6} = \frac{2}{3} \]
Thus, we have:
\[ 7^{1/2 + 1/6} = 7^{2/3} \]
Now, we can convert this back to radical notation. The exponent \( \frac{2}{3} \) means the cube root of \( 7^2 \):
\[ 7^{2/3} = \sqrt[3]{7^2} \]
Thus, the final simplified expression in radical notation is:
\[ \sqrt[3]{49} \]
So, the answer is:
\[ \sqrt[3]{49} \]