Introduction

To prove that the expression (cube root 2 + root 5) * (cube root 2 - root 5) is a rational number, we can simplify the expression using the difference of squares formula and then show that the result is a rational number.

Simplifying the expression

Using the difference of squares formula, we have:

(cube root 2 + root 5) * (cube root 2 - root 5) = (cube root 2)^2 - (root 5)^2

Simplifying further, we get:

(cube root 2)^2 = 2^(2/3) = (2^(1/3))^2 = 2

(root 5)^2 = 5

Therefore, the expression can be simplified as:

2 - 5 = -3

Proving that -3 is a rational number

To prove that -3 is a rational number, we need to show that it can be expressed as a ratio of two integers.

Let's consider -3 as a fraction:

-3 = -3/1

Since -3 can be expressed as a fraction of two integers (-3 and 1), it is a rational number.

Conclusion

By simplifying the expression (cube root 2 + root 5) * (cube root 2 - root 5) using the difference of squares formula, we obtained the result -3. We then proved that -3 is a rational number by showing that it can be expressed as a ratio of two integers (-3/1). Therefore, we have proven that (cube root 2 + root 5) * (cube root 2 - root 5) is a rational number.

In first line cube root 2 +root5