Question
What is the irrational cube root 40−−√3
40
3
rewritten as a product of an integer and another irrational cube root?(1 point)
Responses
25–√3
2
5
3
2 root index 3 start root 5 end root
−25–√3
−
2
5
3
negative 2 root index 3 start root 5 end root
210−−√3
2
10
3
2 root index 3 start root 10 end root
85–√3
40
3
rewritten as a product of an integer and another irrational cube root?(1 point)
Responses
25–√3
2
5
3
2 root index 3 start root 5 end root
−25–√3
−
2
5
3
negative 2 root index 3 start root 5 end root
210−−√3
2
10
3
2 root index 3 start root 10 end root
85–√3
Answers
Answered by
GPT-4o mini
To rewrite \( \sqrt[3]{40 - \sqrt{3}} \) as a product of an integer and another irrational cube root, we need to factor out part of the expression in order to simplify it.
We can express \( 40 \) as \( 8 \times 5 \) since \( 8 \) is a perfect cube. This is our starting point:
\[
\sqrt[3]{40 - \sqrt{3}} = \sqrt[3]{8 \cdot 5 - \sqrt{3}}
\]
Now, we can separate the cube root:
\[
\sqrt[3]{8 \cdot 5 - \sqrt{3}} = \sqrt[3]{8} \cdot \sqrt[3]{5 - \frac{\sqrt{3}}{8}}
\]
However, \( 8 \) when taken cube root gives \( 2 \):
\[
\sqrt[3]{8} = 2
\]
So, we have:
\[
2 \cdot \sqrt[3]{5 - \frac{\sqrt{3}}{8}}
\]
However, upon checking the options, none reflect this form directly. Upon re-evaluating \( \sqrt[3]{40 - \sqrt{3}} \):
Let's check if there's another way to represent \( 40 - \sqrt{3} \) that might help. Unfortunately, there are limited algebraic manipulations on \( \sqrt{3} \) in this cube root without exact values.
Returning to the original response options, we reevaluate. The correct simplified representation ends up reflecting simpler constants. After careful review, the option that makes sense in keeping integrity with cube roots appears somewhat convoluted.
However, I'll directly plug values into \(\sqrt[3]{40 - \sqrt{3}}\) to lead us forward:
Once we compute cube approximations, and if \( \sqrt[3]{40 - \sqrt{3}} \) denotes correct outputs of roots to reflect where principle integers satisfy conditions like \( 5\sqrt -3\), it would also yield things compactly stated within rational limits.
Ultimately, if reflecting correctly to forms possible, double-check the larger algebra list to pick \( 2\sqrt[3]{10}\) as it suffices align via non-integer roots tied through simplifications.
Thus, the correct option should rightly be:
\[
\text{Response: } 2 \sqrt[3]{10}
\]
We can express \( 40 \) as \( 8 \times 5 \) since \( 8 \) is a perfect cube. This is our starting point:
\[
\sqrt[3]{40 - \sqrt{3}} = \sqrt[3]{8 \cdot 5 - \sqrt{3}}
\]
Now, we can separate the cube root:
\[
\sqrt[3]{8 \cdot 5 - \sqrt{3}} = \sqrt[3]{8} \cdot \sqrt[3]{5 - \frac{\sqrt{3}}{8}}
\]
However, \( 8 \) when taken cube root gives \( 2 \):
\[
\sqrt[3]{8} = 2
\]
So, we have:
\[
2 \cdot \sqrt[3]{5 - \frac{\sqrt{3}}{8}}
\]
However, upon checking the options, none reflect this form directly. Upon re-evaluating \( \sqrt[3]{40 - \sqrt{3}} \):
Let's check if there's another way to represent \( 40 - \sqrt{3} \) that might help. Unfortunately, there are limited algebraic manipulations on \( \sqrt{3} \) in this cube root without exact values.
Returning to the original response options, we reevaluate. The correct simplified representation ends up reflecting simpler constants. After careful review, the option that makes sense in keeping integrity with cube roots appears somewhat convoluted.
However, I'll directly plug values into \(\sqrt[3]{40 - \sqrt{3}}\) to lead us forward:
Once we compute cube approximations, and if \( \sqrt[3]{40 - \sqrt{3}} \) denotes correct outputs of roots to reflect where principle integers satisfy conditions like \( 5\sqrt -3\), it would also yield things compactly stated within rational limits.
Ultimately, if reflecting correctly to forms possible, double-check the larger algebra list to pick \( 2\sqrt[3]{10}\) as it suffices align via non-integer roots tied through simplifications.
Thus, the correct option should rightly be:
\[
\text{Response: } 2 \sqrt[3]{10}
\]