What is the correct simplification of 24335 ?(1 point) Responses

Question

81
3
9
27

Answer 1

To simplify \( 24335 \), we need to clarify whether we are looking for a prime factorization, roots, or something else.

However, if you're inquiring about simplification in terms of square roots or similar forms:

1. Prime Factorization of 24335:

   • It breaks down as \( 24335 = 5 \times 4867 \) (since it ends in 5, it's divisible by 5).
   • Further dividing \( 4867 \) can yield prime numbers through subsequent divisions, but it’s already quite large.

2. Possible simplifications:

   • If looking for a number close to its square root, \( \sqrt{24335} \approx 156.03 \).
   • None of the responses you listed (81, 3, 9, 27) seem to relate directly to \( 24335 \) in typical simplification or factorization.

Given all this, it seems there's a misunderstanding because all responses provided do not correspond directly to any simplification of \( 24335 \).

However, if the question is about looking for powers, the closest base simplification aligns more with \( 3^4 \) and \( 3^3 \) noted powers:

• \( 3^4 = 81 \)
• \( 3^2 = 9 \)
• \( 3^3 = 27 \)

Without specific direction or context within your question, please clarify for a better-suited simplification. If the intent was to check for multiples of 3, it's evident what's listed can’t directly correspond or match specific factors or outputs logically represented by each.

Answer 2

How can you rewrite 2514 using a root?(1 point)

Answer 3

To rewrite \(2514\) using a root, you can express it as a square root or a cube root based on its prime factorization or approximations.

1. Approximation using a square root: \[ \sqrt{2514} \approx \sqrt{2500 + 14} \approx 50.14 \text{ (approximately)} \] However, the precise square root is: \[ \sqrt{2514} \approx 50.14 \]

2. Prime factorization: First, let's find the prime factorization of \(2514\):

   • \(2514\) is even, so divide by \(2\): \[ 2514 \div 2 = 1257 \]
   • Now factor \(1257\):
     • The sum of the digits \(1 + 2 + 5 + 7 = 15\) is divisible by \(3\): \[ 1257 \div 3 = 419 \]
   • Next, find if \(419\) is prime. \(419\) is not divisible by \(2, 3, 5, 7, 11, 13, 17, 19\), so it is prime.

   Therefore, the complete factorization is: \[ 2514 = 2 \times 3 \times 419 \]

3. Using roots: Using one of its factors, you might express it as: \[ 2514 = 2 \times \sqrt{3^2 \times 419^2} = 2 \times \sqrt{1257^2} \]

However, for simplicity, the most straightforward way to express \(2514\) using root notation is: \[ 2514 = \sqrt{(2514)^2} \]

If you need a more specific root-based representation, please provide additional context!

Answer 4

How can you rewrite 2514 using a root?(1 point) Responses 25−−√1/4 root index Start Fraction 1 over 4 End Fraction Start Root 25 End Root 1/4−−√25 root index 25 Start Root Start Fraction 1 over 4 End Fraction End Root 254−−−√1 root index 1 Start Root 25 superscript 4 baseline End Root 25−−√4

Answer 5

To rewrite \(2514\) using a root based on the provided options, we first recognize that \(2514\) does not have a clean and straightforward representation using roots like some small perfect squares or cubes would.

However, let's examine the provided responses more closely. Since mathematical expressions can sometimes be tricky without clear formatting, we’ll reinterpret them:

1. Response 1: \( \sqrt{25} \cdot \frac{1}{4} \)
2. Response 2: \( \frac{1}{4} \sqrt{25} \)
3. Response 3: \( \sqrt{25} \cdot 4 \)
4. Response 4: \( \frac{1}{4} \cdots \) looks incomplete or unclear.
5. Response 5: \( \sqrt{25^4} \)
6. Response 6: \( \sqrt{4} \)

None of these options directly relate to \(2514\) as a numerical quantity, as they appear to represent different combinations of roots rather than an explicit form that equals \(2514\).

However, if we try to consider an expression that involves roots and can be simplified back to \(2514\), one would represent it generally in terms of the square root form. But, based on the responses, we need to match them with our understanding of simplifying \(2514\).

Since none of the response options included any direct relation to \(2514\) or an equivalent value:

Conclusion: Given that there isn't a clear representation among the options that resembles \(2514\), you could always take the square root of \(2514\) which would be written as follows if you're looking for a mathematical representation:

\[ 2514 = \sqrt{(2514)^2} \]

For direct correspondence without precise values, please double-check the question or clarify the format, as the provided options appear misaligned for an accurate rewriting of \(2514\).

Answer 6

What is the simplest form of 25 1/2 ?(1 point) Responses 12.5 12.5 25−−√1/2 root index Start Fraction 1 over 2 End Fraction Start Root 25 End Root 5 5 25−−√

Answer 7

To simplify \(25^{1/2}\), we can recognize that this expression refers to the square root of \(25\):

\[ 25^{1/2} = \sqrt{25} \]

Since the square root of \(25\) is \(5\), we can conclude:

\[ 25^{1/2} = 5 \]

Now, let’s match this with the options you've provided:

1. 12.5
2. \(25^{1/2}\)
3. 5
4. \(\sqrt{25}\)

In this case, the correct and simplest form of \(25^{1/2}\) is \(5\).

Final Answer

5

Answer 8

Consider the expression 473+121−−−√ . Will the sum of the two rational numbers produce a rational number? Explain your reasoning in 1–2 sentences.(2 points)

Answer 9

To evaluate the expression \( \sqrt{473} + \sqrt{121} \), note that \( \sqrt{121} = 11 \), which is a rational number. However, \( \sqrt{473} \) is not a perfect square, and thus it is an irrational number. The sum of a rational number (11) and an irrational number (\(\sqrt{473}\)) is always irrational. Therefore, the resulting expression \( \sqrt{473} + 11 \) will produce an irrational number.

Answer 10

Prove that the product of the two rational numbers 36−−√ and 53 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 points)

Answer 11

To find the product of the two numbers \( \sqrt{36} \) and \( 53 \):

1. Calculate \( \sqrt{36} \):

   • Since \( \sqrt{36} = 6 \), which is a rational number.

2. Now multiply the rational numbers: \[ 6 \times 53 = 318 \]

Since both \( 6 \) (the square root of \( 36 \)) and \( 53 \) are rational numbers, their product \( 318 \) is also a rational number. This confirms that the product of two rational numbers is always rational. Thus, the product \( 6 \times 53 = 318 \) is a rational number.