Question

Use radical notation to rewrite the following expression.​ Simplify, if possible.
left parenthesis negative 1024 right parenthesis Superscript two fifths
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Part 1
Rewrite the expression using radical notation.
left parenthesis negative 1024 right parenthesis Superscript two fifths
equals
  
RootIndex 5 StartRoot negative 1024 squared EndRoot
​(Do not simplify. Type an exact​ answer, using radicals as​ needed.)
Part 2
Now simplify.
left parenthesis negative 1024 right parenthesis Superscript two fifths
equals
  
enter your response here
​(Simplify your​ answer.)

Answer 1

To rewrite the expression \(( -1024 )^{ \frac{2}{5} }\) using radical notation:

### Part 1:
The expression can be rewritten as:
\[
( -1024 )^{ \frac{2}{5} } = \sqrt[5]{(-1024)^2}
\]

### Part 2:
To simplify \(( -1024 )^{ \frac{2}{5} }\):
1. Start by calculating \( (-1024)^2 \):
\[
(-1024)^2 = 1024^2 = 1048576
\]

2. Now find the fifth root of \(1048576\):
\[
\sqrt[5]{1048576}
\]

To find the fifth root, we can express \(1048576\) using prime factorization:
\[
1048576 = 2^{20} \quad \text{(since } 1024 = 2^{10}\text{)}
\]

Thus,
\[
\sqrt[5]{1048576} = \sqrt[5]{2^{20}} = 2^{20/5} = 2^4 = 16
\]

Therefore, the simplified expression is:
\[
( -1024 )^{ \frac{2}{5} } = 16
\]

### Final Answer:
1. In radical notation:
\[
( -1024 )^{ \frac{2}{5} } = \sqrt[5]{(-1024)^2}
\]
2. Simplified form:
\[
( -1024 )^{ \frac{2}{5} } = 16
\]