Question

What is the irrational cube root 40−−√3 rewritten as a product of an integer and another irrational cube root?(1 point)
Responses

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

85–√3
8 root index 3 start root 5 end root

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

25–√3

Answer 1

To rewrite \( 40 - \sqrt{3} \) as a product of an integer and another irrational cube root, we first analyze the given expressions:

We want to factor \( 40 \) into a form that allows extraction of a cube root while leaving some component in cube root format.

1. Notice \( 40 \) can be expressed as \( 8 \times 5 \).
2. Also, since we know \( 8 = 2^3 \), we can rewrite \( \sqrt[3]{40} \) as \( \sqrt[3]{8 \cdot 5} \).

So, we have:

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

However, this isn't in a clean factored form yet.

Revisiting the expressions provided:

- \( 2\sqrt[3]{10} \)
- \( 8\sqrt[3]{5} \)
- \( -2\sqrt[3]{5} \)
- \( 25-\sqrt{3} \)

Let's compute rationalizing \( 40 - \sqrt{3} \):

The correct expression that matches our reformulation is:

\[
2\sqrt[3]{5} - \sqrt{3}
\]

However, the closest integer-component and cube-root expression in the options provided needs to notably fit into \( \sqrt[3]{(\text{something})}\).

Let's adjust our terms into the only matching phrase or notable form focused toward the options:

We find that:

- Let's stick with \( 8\sqrt[3]{5} \) leading to:

Thus, analyzing through patterns discernibly:

#### The correct rewritten form of \( 40 - \sqrt{3} \) can be best represented as:
\[
8\sqrt[3]{5}
\]

So, from the given responses, the most appropriate match relating to the adjustments for irrational components can lead toward \( 8\sqrt[3]{5} \) aligns with choice responses.

Thus confirming that obstacles tie-in towards it leading effectively through \(8\sqrt[3]{5}\).