Certainly! Let's explore the problem step by step.
1. Identify the problem: We need to find an example where the addition of two irrational numbers results in a rational number.
2. Understand the properties of irrational and rational numbers:
- An irrational number cannot be expressed as a simple fraction, and its decimal form is non-terminating and non-repeating.
- A rational number can be expressed as a fraction where both the numerator and the denominator are integers.
3. Evaluate each option to see if it can represent the sum of two irrational numbers resulting in a rational number:
### Option 1: [tex]\(\sqrt{7} \cdot \sqrt{3}\)[/tex]
- This represents the product of two irrational numbers, not the sum.
- We need the sum, not the product.
### Option 2: [tex]\(\sqrt{5} \cdot \sqrt{5} = 5\)[/tex]
- This is the product of two identical irrational numbers and it results in a rational number (5).
- However, it is not a sum.
### Option 3: [tex]\(\sqrt{16} \cdot \sqrt{9} = 12\)[/tex]
- This is again the product of two irrational numbers (although, in this case, [tex]\(\sqrt{16} = 4\)[/tex] and [tex]\(\sqrt{9} = 3\)[/tex] are rational numbers).
- But again, it is not a sum.
### Option 4: [tex]\(\pi \cdot \pi = 9.87\)[/tex]
- This is the product of the irrational number [tex]\(\pi\)[/tex] with itself resulting in an approximately irrational number.
- However, it's still about products and not sums.
4. Consider the correct solution example: To find an example where the addition of two irrational numbers results in a rational number, consider adding [tex]\(\sqrt{2}\)[/tex] and [tex]\(-\sqrt{2}\)[/tex]:
[tex]\[ \sqrt{2} + (-\sqrt{2}) \][/tex]
#### Step-by-Step Solution:
- [tex]\(\sqrt{2}\)[/tex] is an irrational number.
- [tex]\(-\sqrt{2}\)[/tex] is also an irrational number because multiplying an irrational number by a rational number (here, [tex]\(-1\)[/tex]) keeps it irrational.
- When we add these two numbers:
[tex]\[ \sqrt{2} + (-\sqrt{2}) = 0 \][/tex]
- The result, 0, is a rational number because it can be expressed as [tex]\( 0/1 \)[/tex].
Therefore, the correct solution is:
[tex]\[ \sqrt{2} + (-\sqrt{2}) = 0 \][/tex]
Example where the addition of two irrational numbers results in a rational number: [tex]\(\sqrt{2}\)[/tex] and [tex]\(-\sqrt{2}\)[/tex]. The sum is [tex]\(0\)[/tex], which is a rational number.
