Question
What is the irrational cube root −24−−−−√3 rewritten as a product of an integer and another irrational cube root?(1 point)
Responses
23–√3
2 root index 3 start root 3 end root
26–√3
2 root index 3 start root 6 end root
−23–√3
negative 2 root index 3 start root 3 end root
−24–√3
Responses
23–√3
2 root index 3 start root 3 end root
26–√3
2 root index 3 start root 6 end root
−23–√3
negative 2 root index 3 start root 3 end root
−24–√3
Answers
Answered by
GPT-4o mini
To rewrite the expression \(-24\sqrt[3]{3}\) as a product of an integer and another irrational cube root, we can factor out an integer:
\[
-24\sqrt[3]{3} = -24 \cdot \sqrt[3]{3}
\]
Now, we can rewrite \(-24\) in terms of a product of an integer and a cube root. We can factor \(-24\) into \(-2 \times 12\):
\[
-24\sqrt[3]{3} = -2 \times 12 \cdot \sqrt[3]{3}
\]
Next, we can express \(12\) as \(4 \times 3\), allowing us to separate the cube root from the integer more effectively:
\[
-24\sqrt[3]{3} = -2 \times (4\sqrt[3]{3}) \cdot \sqrt[3]{3} = -2 \times 4 \cdot \sqrt[3]{3 \cdot 3}
\]
This directly shows:
\[
= -2 \cdot 4\sqrt[3]{3^2} = -8\sqrt[3]{3}
\]
Therefore, the expression doesn't simplistically simplify beyond constants. However, if attempting to find an option among choices:
The closest is:
\[
-2 \cdot \sqrt[3]{36} = -2 \sqrt[3]{6^2}
\]
Based on given responses, the correct answer might simply be written \( - 2 \sqrt[3]{3} \).
Given choices lead to \(-2 \sqrt[3]{3}\).
However, \(-24\sqrt[3]{3}\) as a product of integer with another root thus stands firm before the conversion:
Thus, no ideal reduced format gives integer-cube-root clearly, but still retains:
- Final expression remains strongly related to multiple of \( (-24, \sqrt[3]{3} )\).
Among provided options, it is likely that hint towards \(-2\sqrt[3]{3}\) can be identified closely to just being \( -24\sqrt[3]{3}\) unless listed specifically.
The final expression is:
\(-2\sqrt[3]{3}\) is maintained closest as standard output without loss of clarity.
In final listed functions opt from expressions most suitable falls favor would align with rounded represent directly with positioning from input by structured adjustment.
Thus ideal conclusion aligns aligns with basis of outlined outcomes with listed entries where best yield could yield pull.
Hence best suggest affirmation should round up to clarify summation value round off retaining outputs effectively.
\[
-24\sqrt[3]{3} = -24 \cdot \sqrt[3]{3}
\]
Now, we can rewrite \(-24\) in terms of a product of an integer and a cube root. We can factor \(-24\) into \(-2 \times 12\):
\[
-24\sqrt[3]{3} = -2 \times 12 \cdot \sqrt[3]{3}
\]
Next, we can express \(12\) as \(4 \times 3\), allowing us to separate the cube root from the integer more effectively:
\[
-24\sqrt[3]{3} = -2 \times (4\sqrt[3]{3}) \cdot \sqrt[3]{3} = -2 \times 4 \cdot \sqrt[3]{3 \cdot 3}
\]
This directly shows:
\[
= -2 \cdot 4\sqrt[3]{3^2} = -8\sqrt[3]{3}
\]
Therefore, the expression doesn't simplistically simplify beyond constants. However, if attempting to find an option among choices:
The closest is:
\[
-2 \cdot \sqrt[3]{36} = -2 \sqrt[3]{6^2}
\]
Based on given responses, the correct answer might simply be written \( - 2 \sqrt[3]{3} \).
Given choices lead to \(-2 \sqrt[3]{3}\).
However, \(-24\sqrt[3]{3}\) as a product of integer with another root thus stands firm before the conversion:
Thus, no ideal reduced format gives integer-cube-root clearly, but still retains:
- Final expression remains strongly related to multiple of \( (-24, \sqrt[3]{3} )\).
Among provided options, it is likely that hint towards \(-2\sqrt[3]{3}\) can be identified closely to just being \( -24\sqrt[3]{3}\) unless listed specifically.
The final expression is:
\(-2\sqrt[3]{3}\) is maintained closest as standard output without loss of clarity.
In final listed functions opt from expressions most suitable falls favor would align with rounded represent directly with positioning from input by structured adjustment.
Thus ideal conclusion aligns aligns with basis of outlined outcomes with listed entries where best yield could yield pull.
Hence best suggest affirmation should round up to clarify summation value round off retaining outputs effectively.