Question

the number w is irrational. which statement about \\(\sqrt{5} + w\\) is true?\\(\sqrt{5} + w\\) is rational.\\(\sqrt{5} + w\\) is irrational.\\(\sqrt{5} + w\\) can be rational or irrational, depending on the value of w.

Explanation:

Step1: Recall properties of irrational numbers

An irrational number is a number that cannot be expressed as a fraction of two integers. The sum of an irrational number and another irrational number: Let's assume, for contradiction, that \(\sqrt{5}+w\) is rational. Let \(\sqrt{5}+w = r\), where \(r\) is rational. Then \(w=r - \sqrt{5}\). But \(\sqrt{5}\) is irrational and \(r\) is rational. The difference of a rational and an irrational number is irrational. But we know \(w\) is irrational, so our assumption that \(\sqrt{5}+w\) is rational leads to a consistent result (since \(w\) is irrational). Wait, actually, let's think again. \(\sqrt{5}\) is irrational. If we add two irrational numbers, is the result always irrational? No, for example, \(\sqrt{2}+(-\sqrt{2}) = 0\) (rational). But in our case, \(w\) is irrational, and we are adding it to \(\sqrt{5}\) (irrational). Wait, but the question is: \(w\) is irrational, which statement about \(\sqrt{5}+w\) is true. Let's check the options:

Option 1: \(\sqrt{5}+w\) is rational. Suppose \(\sqrt{5}+w = q\) (rational). Then \(w=q - \sqrt{5}\). Since \(q\) is rational and \(\sqrt{5}\) is irrational, \(q-\sqrt{5}\) is irrational (because rational - irrational = irrational). But \(w\) is given as irrational, so this is possible? Wait, no, the question is which statement is true. Wait, let's take an example. Let \(w = -\sqrt{5}\) (irrational). Then \(\sqrt{5}+w=\sqrt{5}+(-\sqrt{5}) = 0\) (rational). If \(w=\sqrt{2}\) (irrational), then \(\sqrt{5}+\sqrt{2}\) is irrational. Wait, so depending on the value of \(w\), \(\sqrt{5}+w\) can be rational or irrational? But wait, the first option says it's rational. But we saw that when \(w = -\sqrt{5}\), it's rational, but when \(w=\sqrt{2}\), it's irrational. Wait, but the second option says it's irrational. But that's not true. Wait, maybe I made a mistake. Wait, \(\sqrt{5}\) is irrational. Let's recall the theorem: The sum of a rational number and an irrational number is irrational. But here, both \(\sqrt{5}\) and \(w\) are irrational. So the sum of two irrationals can be rational or irrational. So the third option says "can be rational or irrational, depending on \(w\)". But wait, let's check the problem again. The number \(w\) is irrational. Which statement about \(\sqrt{5}+w\) is true.

Wait, let's analyze each option:

1. \(\sqrt{5}+w\) is rational: Is this always true? No, because if \(w=\sqrt{2}\), then \(\sqrt{5}+\sqrt{2}\) is irrational. So this is false.

2. \(\sqrt{5}+w\) is irrational: Is this always true? No, because if \(w = -\sqrt{5}\), then \(\sqrt{5}+w = 0\) (rational). So this is false.

3. \(\sqrt{5}+w\) can be rational or irrational, depending on \(w\). Let's verify:

- Case 1: Let \(w = -\sqrt{5}\) (irrational). Then \(\sqrt{5}+w=\sqrt{5}+(-\sqrt{5}) = 0\) (rational).

- Case 2: Let \(w=\sqrt{2}\) (irrational). Then \(\sqrt{5}+\sqrt{2}\) is irrational (since the sum of two distinct square roots of non - perfect squares is irrational).

So depending on the value of \(w\), the sum can be rational or irrational. Wait, but earlier I thought maybe the second option is correct, but that's not the case. Wait, maybe I messed up the first analysis. Let's re - express:

The key is: when adding two irrational numbers, the result can be rational or irrational. Since \(w\) is irrational and \(\sqrt{5}\) is irrational, their sum can be rational (if \(w\) is the negative of \(\sqrt{5}\)) or irrational (if \(w\) is another irrational number not equal to \(-\sqrt{5}\)). So the third option is correct? Wait, but wait, the first option says it's rational. But we have a case where it's rational (when \(w = -\sqrt{5}\)) and a case where it's irrational (when \(w=\sqrt{2}\)). So the third option says "can be rational or irrational, depending on \(w\)". But let's check the problem again. The number \(w\) is irrational. Which statement is true.

Wait, maybe my initial thought was wrong. Let's recall the property: The sum of an irrational number and a rational number is irrational. But here, both are irrational. So the sum of two irrationals can be rational or irrational. For example:

- \(\sqrt{2}+(-\sqrt{2}) = 0\) (rational, sum of two irrationals)

- \(\sqrt{2}+\sqrt{3}\) (irrational, sum of two irrationals)

So in our problem, \(\sqrt{5}\) is irrational, \(w\) is irrational. So their sum can be rational (if \(w = -\sqrt{5}\)) or irrational (if \(w=\sqrt{2}\)). Therefore, the statement "\(\sqrt{5}+w\) can be rational or irrational, depending on the value of \(w\)" is true. Wait, but let's check the options again:

Option 1: \(\sqrt{5}+w\) is rational. But we saw that when \(w = \sqrt{2}\), it's irrational, so this is false.

Option 2: \(\sqrt{5}+w\) is irrational. But when \(w=-\sqrt{5}\), it's rational, so this is false.

Option 3: \(\sqrt{5}+w\) can be rational or irrational, depending on \(w\). This is true, because we can find values of \(w\) (irrational) that make the sum rational (e.g., \(w = -\sqrt{5}\)) and values that make it irrational (e.g., \(w=\sqrt{2}\)).

Wait, but I think I made a mistake earlier. Let's re - evaluate. The problem says "the number \(w\) is irrational". So we have to see which statement is always true, or which is possible.

Wait, the first option says "is rational" – but it's not always rational. The second says "is irrational" – not always. The third says "can be... depending on \(w\)" – which is true, because there exist irrational \(w\) that make the sum rational (like \(w = -\sqrt{5}\)) and irrational \(w\) that make the sum irrational (like \(w=\sqrt{2}\)).

But wait, let's check the problem again. The options are:

1. \(\sqrt{5}+w\) is rational.

2. \(\sqrt{5}+w\) is irrational.

3. \(\sqrt{5}+w\) can be rational or irrational, depending on \(w\).

So the correct answer is the third option? Wait, but let's think again. Suppose we assume that \(\sqrt{5}+w\) is rational. Then \(w=\text{rational}-\sqrt{5}\). Since \(\sqrt{5}\) is irrational, \(\text{rational}-\sqrt{5}\) is irrational (because rational - irrational = irrational). But \(w\) is given as irrational, so this is consistent. But also, if we take \(w\) as another irrational number, the sum can be irrational. So the sum can be either rational or irrational, depending on \(w\). So the third option is correct.

Wait, but I think I was wrong earlier. Let's take \(w = 1-\sqrt{5}\) (which is irrational, since \(1\) is rational and \(\sqrt{5}\) is irrational, so rational - irrational = irrational). Then \(\sqrt{5}+w=\sqrt{5}+1 - \sqrt{5}=1\) (rational). If \(w=\sqrt{3}\) (irrational), then \(\sqrt{5}+\sqrt{3}\) is irrational. So yes, depending on \(w\), the sum can be rational or irrational. So the third option is correct.

Answer:

\(\boldsymbol{\sqrt{5}+w}\) can be rational or irrational, depending on the value of \(w\). (The option: \(\sqrt{5}+w\) can be rational or irrational, depending on the value of \(w\).)