Write each of the following expressions in simplified exponential form.

Question

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

Answer 1

To convert the given expressions into simplified exponential form, we can use the property of radicals, which states that \(\sqrt[n]{x} = x^{1/n}\).

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. \(\sqrt[15]{27y^{12}}\) \[ \sqrt[15]{27y^{12}} = 27^{1/15}y^{12/15} = 3^{3/15}y^{4/5} = 3^{1/5}y^{4/5} \]

Part 5

e. \(27\sqrt[15]{y^{12}}\) \[ 27\sqrt[15]{y^{12}} = 27 \cdot y^{12/15} = 27 \cdot y^{4/5} = 3^3 \cdot y^{4/5} = 3^3y^{4/5} \]

Final Responses

• 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: \(3^3y^{4/5}\)