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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
negative 2 root index 3 start root 5 end root

25–√3
2 root index 3 start root 5 end root

210−−√3
2 root index 3 start root 10 end root

85–√3

3 answers

To rewrite \( 40\sqrt[3]{3} \) as a product of an integer and another irrational cube root, we first need to factor \( 40 \) in a way that separates an integer and the remaining part under the cube root.

The number \( 40 \) can be factored as:

\[ 40 = 8 \times 5 \]

Now, we know that \( 8 \) is a perfect cube (since \( 8 = 2^3 \)). Thus, we can rewrite:

\[ 40\sqrt[3]{3} = (8)\sqrt[3]{5}\sqrt[3]{3} \]

This simplifies to:

\[ = 2^3 \cdot \sqrt[3]{15} \] \[ = 2\sqrt[3]{8 \cdot 5 \cdot 3} = 2\sqrt[3]{120} \]

However, we want it in a more suitable form. Since \( 40 = 8 \cdot 5 \), we can rewrite:

\[ 40\sqrt[3]{3} = 2\sqrt[3]{8} \cdot \sqrt[3]{5} \cdot \sqrt[3]{3} = 2 \cdot \sqrt[3]{15 \cdot 3} = 2\sqrt[3]{15} \cdot \sqrt[3]{3} \]

Thus, your expression is presented as:

\[ = 2\sqrt[3]{40} \]

Thus, writing it as the form requested we can say:

The closest option to the derived answer is:

  • \( 2\sqrt[3]{15} \)

However, it's not in your option, since let's review the options presented:

  1. \( -25\sqrt[3]{3} \)
  2. \( 2\sqrt[3]{5} \)
  3. \( 2\sqrt[3]{10} \)
  4. \( 85\sqrt[3]{3} \)

Since none perfectly match \( \sqrt[3]{5} \) we are presented closest to \( 2\sqrt[3]{40} \).

So we take the best fitting which would be:

\[ 2\sqrt[3]{15} \]

If we take the view of forms generally the response seems like \( 2\sqrt[3]{10} \), in this analytical frame remains to reason what would specifically return on insights of \( 2\sqrt[3]{3} \).

Thats not one of the answers

I apologize for the confusion in my previous response. Let's correctly rewrite \( 40\sqrt[3]{3} \) as a product of an integer and another irrational cube root, and then check against the answer choices provided.

To do this:

  1. Factor \( 40 \) into \( 8 \times 5 \), where \( 8 \) is a perfect cube.
  2. We can express \( 40\sqrt[3]{3} \) as:

\[ 40\sqrt[3]{3} = 8 \cdot 5 \cdot \sqrt[3]{3} = 8 \cdot \sqrt[3]{3} \cdot 5 = 8\sqrt[3]{3} \times \sqrt[3]{5} \] \[ = 2^3 \cdot \sqrt[3]{15} \] \[ = 2 \cdot \sqrt[3]{40} \text{ (to form a general form of integer times cube root)} \]

Next, we want this in the form of \( k \sqrt[3]{m} \).

So now looking at the provided answers:

  1. \(-25\sqrt[3]{3}\)
  2. \(2\sqrt[3]{5}\)
  3. \(2\sqrt[3]{10}\)
  4. \(85\sqrt[3]{3}\)

We can see that the form \( 2\sqrt[3]{5} \) closely aligns with the structure while retaining the value scrutiny for rational numbers.

So if \(40 = 8 \cdot 5\) it can become \(2\sqrt[3]{40}\) but does advise closer to an after-simplification. Contemplating remains that:

The right option is:

\( 2\sqrt[3]{10} \).

Please review these associations suggesting \( k\sqrt[3]{m}\) appropriately to the original query, and my sincere apologies for earlier inaccuracies!