Consider the following set of real numbers:


open curly brackets short dash square root of 7 comma space short dash 0.7 with bar on top comma space 1.5 comma space square root of 25 close curly brackets


Which of the following contains all of the irrational numbers in the set?


A.)
short dash square root of 7 comma space square root of 25


B.)
short dash 0.7 with bar on top comma space 1.5 comma space square root of 25


C.)
negative root index blank of 7 comma end root short dash 0.7 with bar on top comma space square root of 25


D.)
short dash square root of 7

To determine which numbers in the given set are **irrational**, we analyze each element:

### Given set:  
\[
\{-\sqrt{7},\ -0.\overline{7},\ 1.5,\ \sqrt{25}\}
\]

#### Step 1: Identify Rational and Irrational Numbers  
1. **\(-\sqrt{7}\)**  
   - The square root of 7 is **irrational** because 7 is not a perfect square.  
   - The negative of an irrational number is still **irrational**.  
   - So, **\(-\sqrt{7}\) is irrational**.

2. **\(-0.\overline{7}\) (Repeating decimal)**  
   - A repeating decimal represents a rational number.  
   - So, **\(-0.\overline{7}\) is rational**.

3. **\(1.5\)**  
   - This is a terminating decimal, which is a **rational number**.  
   - So, **\(1.5\) is rational**.

4. **\(\sqrt{25}\)**  
   - Since \(\sqrt{25} = 5\), which is a whole number, it is **rational**.  
   - So, **\(\sqrt{25}\) is rational**.

#### Step 2: Find the Answer Choice Containing Only Irrational Numbers  
- The only **irrational number** in the set is **\(-\sqrt{7}\)**.  
- The correct answer is:

**D.)** \(-\sqrt{7}\)

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Would you like a more detailed breakdown or a different explanation?  

### Related Questions:
1. What makes a number irrational?
2. How do you convert repeating decimals into fractions?
3. Can a negative number be irrational?
4. How do you determine if a square root is rational or irrational?
5. Why is \(\sqrt{25}\) considered rational?

### **Tip:**  
A number is **irrational** if it **cannot** be written as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q \neq 0\). Examples include non-repeating, non-terminating decimals like \(\pi\) and \(\sqrt{2}\).

Consider the following set of real numbers: {-√7, -0.7̅, 1.5, √25}. Which of the following contains all of the irrational numbers in the set?

A.) -√7, √25
B.) -0.7̅, 1.5, √25
C.) -√7, -0.7̅, √25
D.) -√7

Learn how to identify irrational numbers in a given set of real numbers, including square roots and repeating decimals. This problem covers key mathematical concepts such as rational vs. irrational numbers and is suitable for students in Grades 8-10.

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