To complete the table, we will convert between rational exponent form and root form, and then simplify the expressions.
Table:
| Rational Exponent Form | Root Form | Simplify | |------------------------|-----------|-----------------| | \( 81^{\frac{1}{4}} \) | \( \sqrt[4]{81} \) | \( 3 \) | | \( 121^{\frac{1}{2}} \) | \( \sqrt{121} \) | \( 11 \) |
Explanation:
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For \( 81^{\frac{1}{4}} \):
- Root Form: The fourth root of 81 can be written as \( \sqrt[4]{81} \).
- Simplification: Since \( 81 = 3^4 \), we find \( \sqrt[4]{81} = \sqrt[4]{3^4} = 3 \).
-
For \( 121^{\frac{1}{2}} \):
- Root Form: This is the square root of 121, which can be written as \( \sqrt{121} \).
- Simplification: Since \( 121 = 11^2 \), we have \( \sqrt{121} = 11 \).
Thus, the table is complete with both the rational exponent forms, root forms, and their simplifications filled in.