Consider the expression
5/2+3/2. Will the sum of the two rational numbers produce a rational number? Explain your reasoning in 1-2 sentences.
2 months ago
2 months ago
Prove that the product of the two rational numbers
4/3 and 9/2 is a rational number. Find the product and explain how you know it is a rational number. Explain your reasoning in 1-2 sentences
2 months ago
To find the product of the two rational numbers \( \frac{4}{3} \) and \( \frac{9}{2} \), we multiply them as follows:
\[
\frac{4}{3} \times \frac{9}{2} = \frac{4 \times 9}{3 \times 2} = \frac{36}{6} = 6.
\]
The product \( 6 \) is a rational number because it can be expressed as a ratio of integers (\( \frac{6}{1} \)). This proves that the product of two rational numbers is always a rational number.
2 months ago
Prove that the sum of
1/2 andโ2 is irrational. Show your work and in 1-2 sentences explain why the sum is an irrational number.
2 months ago
To prove that the sum \( \frac{1}{2} + \sqrt{2} \) is irrational, we can use a proof by contradiction.
1. Assume that the sum \( \frac{1}{2} + \sqrt{2} \) is rational. Then, we can express it as \( r \), where \( r \) is a rational number. This gives us:
\[
\sqrt{2} = r - \frac{1}{2}
\]
2. Since \( r \) is rational and \( \frac{1}{2} \) is rational, their difference \( r - \frac{1}{2} \) must also be rational.
3. Therefore, if \( \sqrt{2} \) is rational, this implies that \( \sqrt{2} \) can be expressed as a ratio of two integers. However, it is known that \( \sqrt{2} \) is irrational, as it cannot be expressed as a fraction of two integers without leading to a contradiction (such as assuming it can be written as \( \frac{a}{b} \) in lowest terms and arriving at a contradiction when squaring both sides).
4. Since our assumption that \( \frac{1}{2} + \sqrt{2} \) is rational leads us to a contradiction, we conclude that the assumption must be false. Therefore, the sum \( \frac{1}{2} + \sqrt{2} \) is irrational.
In summary, we know the sum is irrational because it relies on the irrationality of \( \sqrt{2} \); since the result of adding a rational number (like \( \frac{1}{2} \)) to an irrational number (like \( \sqrt{2} \)) is always irrational, the sum \( \frac{1}{2} + \sqrt{2} \) must also be irrational.
2 months ago
Prove that the product of 3ฯ and
2/5 is an irrational number. Find the product and explain why the value is irrational. Explain your reasoning in 1-2 sentences.
2 months ago
To prove that the product \( 3\pi \times \frac{2}{5} \) is irrational, we first compute the product:
\[
3\pi \times \frac{2}{5} = \frac{6\pi}{5}.
\]
Next, we need to show that \( \frac{6\pi}{5} \) is irrational. The number \( \pi \) is known to be an irrational number, meaning it cannot be expressed as a fraction of two integers. If we assume \( \frac{6\pi}{5} \) were rational, we could express it as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers.
This would imply that \( \pi = \frac{5p}{6q} \), which suggests \( \pi \) could be expressed as a ratio of integers, contradicting its known irrationality. Therefore, \( 3\pi \times \frac{2}{5} = \frac{6\pi}{5} \) is irrational.
In summary, the product \( \frac{6\pi}{5} \) is irrational because it is a non-zero multiple of the irrational number \( \pi \), and the product of a non-zero rational number and an irrational number is always irrational.
2 months ago
Describe why simplifying square roots into their most reduced radical form is important in mathematical problem-solving. Also, outline the steps involved in simplifying a square root to its simplest radical form.
2 months ago
### Importance of Simplifying Square Roots
Simplifying square roots into their most reduced radical form is important in mathematical problem-solving for several reasons:
1. **Clarity and Precision**: Simplified radicals are clearer and easier to interpret, which helps in understanding the results and concepts involved in mathematical problems.
2. **Ease of Calculation**: Working with simpler expressions makes it easier to perform further calculations, whether in algebra, geometry, or trigonometry.
3. **Standard Form**: In many mathematical contexts, particularly in higher-level mathematics, itโs expected to present answers in simplest form. This aids in communication and comparability of results.
4. **Error Reduction**: Simplification can help identify and reduce errors in calculations, as simpler forms are less prone to computational complexities or misinterpretation.
### Steps to Simplifying a Square Root to Its Simplest Radical Form
1. **Factor the Radicand**: Identify the number under the square root (the radicand) and factor it into prime factors.
2. **Pair the Factors**: Group the factors into pairs. Each pair of the same number can be taken out of the square root.
3. **Take Out the Pairs**: For every pair of identical factors, take one factor out of the square root.
4. **Multiply the Factors Outside**: Multiply all factors that have been taken outside the square root together. This will be the coefficient in front of the root.
5. **Keep the Remaining Factors Inside**: Any factors that could not be paired will remain inside the square root.
6. **Write the Final Result**: Combine the coefficient with the square root of the remaining factors to express the simplified radical form.
### Example
Let's simplify \( \sqrt{72} \):
1. **Factor the Radicand**: \( 72 = 2^3 \times 3^2 \).
2. **Pair the Factors**: From \( 2^3 \), we can take out one \( 2 \) (since thereโs one pair of 2s), and from \( 3^2 \), we can take out one \( 3 \) (since there's a pair of 3s).
3. **Take Out the Pairs**: \( \sqrt{72} = \sqrt{2^2 \times 2 \times 3^2} = 2 \times 3 \times \sqrt{2} \).
4. **Multiply the Factors Outside**: \( 2 \times 3 = 6 \).
5. **Keep the Remaining Factors Inside**: The remaining \( \sqrt{2} \) stays inside.
6. **Write the Final Result**: Thus, \( \sqrt{72} = 6\sqrt{2} \).
This process results in the final answer in its simplest radical form.
2 months ago
make that shorter in a sentence
2 months ago