Which of the following is irrational?(1 point)

Question

Responses

3.6363636363636363...
3.6363636363636363...

−7837
negative 7 Start Fraction 8 over 37 End Fraction

3√4
Start Fraction Start Root 3 End Root over 4 End Fraction

52.781654292

Answer 1

To determine which of the given responses is irrational, let's analyze each one:

1. 3.6363636363636363...: This is a repeating decimal (specifically \(3.\overline{63}\)), which can be expressed as a fraction. Therefore, it is rational.

2. −7837: This is a negative integer, which is also a rational number since any integer can be expressed as a fraction (e.g., \(-7837/1\)).

3. Negative \(7 \frac{8}{37}\): This is a mixed number, which can be converted to a fraction. The improper fraction form is \(-\frac{7 \cdot 37 + 8}{37} = -\frac{259}{37}\), making it rational as well.

4. \(3\sqrt{4}\): The square root of 4 is 2, so \(3\sqrt{4} = 3 \cdot 2 = 6\), which is an integer and hence rational.

5. 52.781654292: This is a decimal number that terminates. Any terminating decimal can be expressed as a fraction, so it is rational.

Since all the options provided are rational numbers, none of them are irrational according to the typical definitions of rational and irrational numbers.

Thus, if I were to pick a response based on irrationality, none of these options qualify as irrational. However, if your prompt absolutely requires you to identify an answer, the closest misunderstood presentation may be "3√4" due to misinterpretation, but under mathematical definitions, it is still rational. Please confirm if there's additional context or clarification needed!

Answer 2

Estimate the value of 6×2–√ . Which of the following choices is the best estimate?(1 point)
Responses

6.4
6.4

8.4
8.4

7.1
7.1

9.8

Answer 3

To estimate the value of \(6 \times 2 - \sqrt{6}\), we can break it down into parts.

1. Calculate \(6 \times 2\): \[ 6 \times 2 = 12 \]

2. Estimate \(\sqrt{6}\): The square root of 6 can be estimated. Since \(2^2 = 4\) and \(3^2 = 9\), we know that \(\sqrt{6}\) is between 2 and 3. A better estimate would be around 2.45 (since 2.45 squared is approximately 6).

3. Combine the results: Now, subtract our estimate of \(\sqrt{6}\) from \(12\): \[ 12 - \sqrt{6} \approx 12 - 2.45 \approx 9.55 \]

Based on these calculations, the best estimate for \(6 \times 2 - \sqrt{6}\) is approximately 9.5.

Considering the options provided:

• 6.4
• 8.4
• 7.1
• 9.8

The closest estimate to our calculated value (9.55) is 9.8.

Thus, the best estimate is 9.8.

Answer 4

Which statement is true about the square root of 28?(1 point)
Responses

It is between 4 and 5, probably closer to 5.
It is between 4 and 5, probably closer to 5.

It is between 5 and 6, probably closer to 5.
It is between 5 and 6, probably closer to 5.

It is between 6 and 7, probably closer to 6.
It is between 6 and 7, probably closer to 6.

It is between 5 and 6, probably closer to 6.

Answer 5

To determine where the square root of 28 falls, let's calculate it more precisely:

1. Identify perfect squares around 28:
   • \(5^2 = 25\)
   • \(6^2 = 36\)

From this, we can see that: \[ 5^2 < 28 < 6^2 \] Thus, \(\sqrt{28}\) is between 5 and 6.

2. Calculate \(\sqrt{28}\): \(\sqrt{28}\) can be simplified as: \[ \sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7} \]

3. Estimate \(\sqrt{7}\): We know that \( \sqrt{7} \) is between \( 2.6 \) and \( 2.7 \) (since \(2.6^2 = 6.76\) and \(2.7^2 = 7.29\)). For a quick estimate, let's take \( \sqrt{7} \approx 2.645\).

Thus: \[ 2\sqrt{7} \approx 2 \times 2.645 \approx 5.29 \]

4. Conclude the estimate: Since \( \sqrt{28} \approx 5.29\), it is indeed between 5 and 6, and it is closer to 5.

Based on this analysis, the correct statement is:

"It is between 5 and 6, probably closer to 5."