Find the sum: 56−−√3+57–√3
.

A. 67–√3
B. 663−−√3
C. 7–√3
D. 77–√3

To find the sum \( \sqrt[3]{56} + \sqrt[3]{57} \), we need to understand how cube roots behave. Given that 56 and 57 are very close to each other, their cube roots will also be close.

Let's denote:
\[ a = \sqrt[3]{56} \]
\[ b = \sqrt[3]{57} \]

We are asked to find \( a + b \). Notice that both \( a \) and \( b \) are real numbers.

To approximate these cube roots, we use the fact that cube roots of numbers close together are themselves close together. For simplification:

- The cube root of 56 is slightly less than 4 (since \(4^3 = 64\)).
- The cube root of 57 is also slightly less than 4 but more than \( \sqrt[3]{56} \).

Since \( 56 \) and \( 57 \) are consecutive numbers and close to \( 4^3 \) (which is 64), their cube roots are close to \( \sqrt[3]{64} = 4 \).

Approximating:
\[ \sqrt[3]{56} \approx 3.83 \]
\[ \sqrt[3]{57} \approx 3.85 \]

Adding these approximations:
\[ \sqrt[3]{56} + \sqrt[3]{57} \approx 3.83 + 3.85 = 7.68 \]

Given the options and using exact calculations rather than approximations is ideal, but our choices can guide us to confirm:

Given choices are:
A. \( \sqrt[3]{67} \)
B. \( \sqrt[3]{663} \)
C. \( 7 \sqrt[3] \)
D. \( 77 \sqrt[3] \)

Since \( \sqrt[3]{56} \approx 3.83 \) and \( \sqrt[3]{57} \approx 3.85 \), they add up roughly to around 7, closer to option \( C \):

\[ \boxed{7 \sqrt[3]{}} \]

So the correct choice is:

C. \( 7 \sqrt[3] \)

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you approximate the cube root of a number?
2. What is the cube root of 64, and how does it help estimate the cube roots of nearby numbers?
3. How would you compute \( \sqrt[3]{56} \) and \( \sqrt[3]{57} \) more precisely?
4. Can the sum of cube roots be simplified like the sum of square roots?
5. How do you handle sums of higher order roots?

**Tip:** When estimating roots, knowing nearby exact roots (like \( \sqrt[3]{64} = 4 \)) helps approximate values of close numbers.

Learn how to find the sum of cube roots \( \sqrt[3]{56} + \sqrt[3]{57} \) with a detailed explanation. Understand cube root approximation and how to estimate values of cube roots of nearby numbers. Suitable for high school students.

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