Answer :
Let's go through each part of the exercise step-by-step.
### Part 1: Rational and Irrational Numbers
Question: Which of [tex]\(0.3\)[/tex], [tex]\(\pi\)[/tex], [tex]\(\sqrt{25}\)[/tex], and [tex]\(\sqrt{5}\)[/tex] are rational?
Solution:
- [tex]\(0.3\)[/tex]: This is a terminating decimal, which means it can be expressed as a fraction. Specifically, [tex]\(0.3 = \frac{3}{10}\)[/tex]. Therefore, [tex]\(0.3\)[/tex] is a rational number.
- [tex]\(\pi\)[/tex]: Pi ([tex]\(\pi\)[/tex]) is a well-known irrational number. It cannot be expressed as a fraction of two integers.
- [tex]\(\sqrt{25}\)[/tex]: The square root of 25 is 5, which is an integer. Integers are rational numbers (since they can be expressed as fractions where the denominator is 1), so [tex]\(\sqrt{25}\)[/tex] is rational.
- [tex]\(\sqrt{5}\)[/tex]: The square root of 5 cannot be expressed as a fraction. Therefore, [tex]\(\sqrt{5}\)[/tex] is an irrational number.
So, the rational numbers among [tex]\(0.3\)[/tex], [tex]\(\pi\)[/tex], [tex]\(\sqrt{25}\)[/tex], and [tex]\(\sqrt{5}\)[/tex] are [tex]\(0.3\)[/tex] and [tex]\(\sqrt{25}\)[/tex].
### Part 2: Finding a Rational Number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex]
Question: Find a rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex].
Solution:
- The square root of 3 is approximately 1.732.
- The square root of 5 is approximately 2.236.
A rational number between these two values is their average. So, we calculate:
[tex]\[ \text{Average} = \frac{\sqrt{3} + \sqrt{5}}{2} \][/tex]
Substituting approximate values:
[tex]\[ \text{Average} = \frac{1.732 + 2.236}{2} \approx \frac{3.968}{2} \approx 1.984 \][/tex]
Therefore, a rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex] is approximately 1.984.
### Part 3: Finding an Irrational Number between 3 and 4
Question: Find an irrational number between 3 and 4.
Solution:
- The square root of 10 is approximately [tex]\(\sqrt{10} \approx 3.162\)[/tex].
Since [tex]\(\sqrt{10}\)[/tex] is an irrational number and its value lies between 3 and 4, we can use:
[tex]\[ \sqrt{10} \approx 3.162 \][/tex]
As an example of an irrational number between 3 and 4.
### Part 4: Writing [tex]\(3 \sqrt{5}\)[/tex] as the Square Root of a Single Number
Question: Write [tex]\(3 \sqrt{5}\)[/tex] as the square root of a single number.
Solution:
We have:
[tex]\[ 3 \sqrt{5} \][/tex]
One way to express this in the form of a single square root is to first square the expression:
[tex]\[ (3 \sqrt{5})^2 = 9 \times 5 = 45 \][/tex]
Now, we take the square root of 45:
[tex]\[ \sqrt{45} = 3 \sqrt{5} \][/tex]
Therefore, [tex]\(3 \sqrt{5}\)[/tex] can be written as:
[tex]\[ \sqrt{45} \][/tex]
So, [tex]\(3 \sqrt{5}\)[/tex] equals [tex]\(\sqrt{45}\)[/tex].
### Summary
1. The rational numbers among [tex]\(0.3\)[/tex], [tex]\(\pi\)[/tex], [tex]\(\sqrt{25}\)[/tex], and [tex]\(\sqrt{5}\)[/tex] are [tex]\(0.3\)[/tex] and [tex]\(\sqrt{25}\)[/tex].
2. A rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex] is approximately 1.984.
3. An irrational number between 3 and 4 is [tex]\(\sqrt{10}\)[/tex].
4. [tex]\(3 \sqrt{5}\)[/tex] can be written as [tex]\(\sqrt{45}\)[/tex].
### Part 1: Rational and Irrational Numbers
Question: Which of [tex]\(0.3\)[/tex], [tex]\(\pi\)[/tex], [tex]\(\sqrt{25}\)[/tex], and [tex]\(\sqrt{5}\)[/tex] are rational?
Solution:
- [tex]\(0.3\)[/tex]: This is a terminating decimal, which means it can be expressed as a fraction. Specifically, [tex]\(0.3 = \frac{3}{10}\)[/tex]. Therefore, [tex]\(0.3\)[/tex] is a rational number.
- [tex]\(\pi\)[/tex]: Pi ([tex]\(\pi\)[/tex]) is a well-known irrational number. It cannot be expressed as a fraction of two integers.
- [tex]\(\sqrt{25}\)[/tex]: The square root of 25 is 5, which is an integer. Integers are rational numbers (since they can be expressed as fractions where the denominator is 1), so [tex]\(\sqrt{25}\)[/tex] is rational.
- [tex]\(\sqrt{5}\)[/tex]: The square root of 5 cannot be expressed as a fraction. Therefore, [tex]\(\sqrt{5}\)[/tex] is an irrational number.
So, the rational numbers among [tex]\(0.3\)[/tex], [tex]\(\pi\)[/tex], [tex]\(\sqrt{25}\)[/tex], and [tex]\(\sqrt{5}\)[/tex] are [tex]\(0.3\)[/tex] and [tex]\(\sqrt{25}\)[/tex].
### Part 2: Finding a Rational Number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex]
Question: Find a rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex].
Solution:
- The square root of 3 is approximately 1.732.
- The square root of 5 is approximately 2.236.
A rational number between these two values is their average. So, we calculate:
[tex]\[ \text{Average} = \frac{\sqrt{3} + \sqrt{5}}{2} \][/tex]
Substituting approximate values:
[tex]\[ \text{Average} = \frac{1.732 + 2.236}{2} \approx \frac{3.968}{2} \approx 1.984 \][/tex]
Therefore, a rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex] is approximately 1.984.
### Part 3: Finding an Irrational Number between 3 and 4
Question: Find an irrational number between 3 and 4.
Solution:
- The square root of 10 is approximately [tex]\(\sqrt{10} \approx 3.162\)[/tex].
Since [tex]\(\sqrt{10}\)[/tex] is an irrational number and its value lies between 3 and 4, we can use:
[tex]\[ \sqrt{10} \approx 3.162 \][/tex]
As an example of an irrational number between 3 and 4.
### Part 4: Writing [tex]\(3 \sqrt{5}\)[/tex] as the Square Root of a Single Number
Question: Write [tex]\(3 \sqrt{5}\)[/tex] as the square root of a single number.
Solution:
We have:
[tex]\[ 3 \sqrt{5} \][/tex]
One way to express this in the form of a single square root is to first square the expression:
[tex]\[ (3 \sqrt{5})^2 = 9 \times 5 = 45 \][/tex]
Now, we take the square root of 45:
[tex]\[ \sqrt{45} = 3 \sqrt{5} \][/tex]
Therefore, [tex]\(3 \sqrt{5}\)[/tex] can be written as:
[tex]\[ \sqrt{45} \][/tex]
So, [tex]\(3 \sqrt{5}\)[/tex] equals [tex]\(\sqrt{45}\)[/tex].
### Summary
1. The rational numbers among [tex]\(0.3\)[/tex], [tex]\(\pi\)[/tex], [tex]\(\sqrt{25}\)[/tex], and [tex]\(\sqrt{5}\)[/tex] are [tex]\(0.3\)[/tex] and [tex]\(\sqrt{25}\)[/tex].
2. A rational number between [tex]\(\sqrt{3}\)[/tex] and [tex]\(\sqrt{5}\)[/tex] is approximately 1.984.
3. An irrational number between 3 and 4 is [tex]\(\sqrt{10}\)[/tex].
4. [tex]\(3 \sqrt{5}\)[/tex] can be written as [tex]\(\sqrt{45}\)[/tex].
