Question

Write each of the following expressions in simplified exponential form.
a. StartRoot x cubed EndRoot
b. RootIndex 7 StartRoot x cubed EndRoot
c. RootIndex 8 StartRoot x cubed EndRoot
d. RootIndex 15 StartRoot 27 y Superscript 12 EndRoot
e. 27 RootIndex 15 StartRoot y Superscript 12 EndRoot
Question content area bottom
Part 1
a. StartRoot x cubed EndRootequals
  
enter your response here
​(Simplify your answer. Type your answer using exponential​ notation.)
Part 2
b. RootIndex 7 StartRoot x cubed EndRootequals
  
enter your response here
​(Simplify your answer. Type your answer using exponential​ notation.)
Part 3
c. RootIndex 8 StartRoot x cubed EndRootequals
  
enter your response here
​(Simplify your answer. Type your answer using exponential​ notation.)
Part 4
d. RootIndex 15 StartRoot 27 y Superscript 12 EndRootequals
  
enter your response here
​(Simplify your answer. Type your answer using exponential​ notation.)
Part 5
e. 27 RootIndex 15 StartRoot y Superscript 12 EndRootequals
  
enter your response here
​(Simplify your answer. Type your answer using exponential​ notation.)

Answer 1

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.