Question
What is the irrational cube root 40−−√3 rewritten as a product of an integer and another irrational cube root?(1 point)
Responses
25–√3
2 root index 3 start root 5 end root
85–√3
8 root index 3 start root 5 end root
210−−√3
2 root index 3 start root 10 end root
−25–√3
Responses
25–√3
2 root index 3 start root 5 end root
85–√3
8 root index 3 start root 5 end root
210−−√3
2 root index 3 start root 10 end root
−25–√3
Answers
Answered by
GPT-4o mini
To rewrite the expression \(40 - \sqrt{3}\) as a product of an integer and another irrational cube root, let's analyze the options you provided.
First, we can factor 40 into \(8 \cdot 5\) (since \(8 = 2^3\)), which could be helpful since we are looking at cube roots:
\[
40 - \sqrt{3} = 8 \cdot 5 - \sqrt{3}
\]
If we want to express it in the form of an integer multiplied by a cube root, we can isolate the \(8\):
\[
= 8 \left(5 - \frac{\sqrt{3}}{8}\right)
\]
Since we are not finding a straightforward way to express \(5 - \frac{\sqrt{3}}{8}\) directly as a cube root, we should review the given options.
From the options, only option 2 looks promising:
\[
8 \sqrt[3]{5}
\]
To confirm the correctness of \(8 \sqrt[3]{5}\), we'll check if \(8 \sqrt[3]{5}\) translates back to our original expression by considering it leads us around in squares of sums or cubes.
None of the options presented initially seem to directly lead us from \(40 - \sqrt{3}\), so upon revisiting and referring to common cube root manipulations, we'll find \(2 \sqrt[3]{10}\) as another candidate since \(10\) is half of \(20\) also taking cube of \(2\).
However, to answer directly based on the image:
Since we are essentially looking for the expression of \(40 - \sqrt{3}\) in form is \(8 \sqrt[3]{5}\).
So, the answer is:
**8 \sqrt[3]{5}**.
First, we can factor 40 into \(8 \cdot 5\) (since \(8 = 2^3\)), which could be helpful since we are looking at cube roots:
\[
40 - \sqrt{3} = 8 \cdot 5 - \sqrt{3}
\]
If we want to express it in the form of an integer multiplied by a cube root, we can isolate the \(8\):
\[
= 8 \left(5 - \frac{\sqrt{3}}{8}\right)
\]
Since we are not finding a straightforward way to express \(5 - \frac{\sqrt{3}}{8}\) directly as a cube root, we should review the given options.
From the options, only option 2 looks promising:
\[
8 \sqrt[3]{5}
\]
To confirm the correctness of \(8 \sqrt[3]{5}\), we'll check if \(8 \sqrt[3]{5}\) translates back to our original expression by considering it leads us around in squares of sums or cubes.
None of the options presented initially seem to directly lead us from \(40 - \sqrt{3}\), so upon revisiting and referring to common cube root manipulations, we'll find \(2 \sqrt[3]{10}\) as another candidate since \(10\) is half of \(20\) also taking cube of \(2\).
However, to answer directly based on the image:
Since we are essentially looking for the expression of \(40 - \sqrt{3}\) in form is \(8 \sqrt[3]{5}\).
So, the answer is:
**8 \sqrt[3]{5}**.