To rewrite each expression in simplified exponential form, we can use the property of exponents that states \(\sqrt[n]{a^m} = a^{m/n}\).
Let's simplify each expression one by one:
Part 1
a. \(\sqrt{x^3}\)
\[ \sqrt{x^3} = x^{3/2} \]
Part 2
b. \(\sqrt[7]{x^3}\)
\[ \sqrt[7]{x^3} = x^{3/7} \]
Part 3
c. \(\sqrt[8]{x^3}\)
\[ \sqrt[8]{x^3} = x^{3/8} \]
Part 4
d. \(\sqrt[15]{27y^{12}}\)
First, express \(27\) as \(3^3\):
\[ \sqrt[15]{27y^{12}} = \sqrt[15]{3^3 \cdot y^{12}} = \sqrt[15]{3^3} \cdot \sqrt[15]{y^{12}} = 3^{3/15} \cdot y^{12/15} = 3^{1/5} \cdot y^{4/5} \]
Part 5
e. \(27\sqrt[15]{y^{12}}\)
Again, express \(27\) as \(3^3\):
\[ 27\sqrt[15]{y^{12}} = 3^3 \cdot \sqrt[15]{y^{12}} = 3^3 \cdot y^{12/15} = 3^3 \cdot y^{4/5} = 27 \cdot y^{4/5} \]
Summary of Answers
- Part 1: \(x^{3/2}\)
- Part 2: \(x^{3/7}\)
- Part 3: \(x^{3/8}\)
- Part 4: \(3^{1/5} y^{4/5}\)
- Part 5: \(27y^{4/5}\)
These are the simplified exponential forms for each expression.