QuestionIdentify the irrational numbers from the list: $\sqrt{36}$ , $\sqrt{23}$ , $-4.232323 \ldots$ , $0.151551555 \ldots$ , $0.3 \overline{5}$ , $\pi$ .

Studdy Solution

STEP 1

Assumptions1. We are given six numbers $\sqrt{36}$ , $\sqrt{23}$ , $-4.232323 \ldots$ , $0.151551555 \ldots$ , $0.3 \overline{5}$ , and $\pi$ . . We need to determine which of these numbers are irrational.
3. A number is irrational if it cannot be expressed as a ratio of two integers. In other words, it cannot be written as a fraction where both the numerator and the denominator are integers.
4. The decimal representation of an irrational number neither terminates nor repeats.

STEP 2

Evaluate each number to determine if it is irrational.
First, let's evaluate $\sqrt{36}$ .

STEP 3

Calculate the square root of36.
$\sqrt{36} =6$

STEP 4

Since6 can be expressed as a ratio of two integers (6/1), $\sqrt{36}$ is not irrational.
Next, let's evaluate $\sqrt{23}$ .

STEP 5

Calculate the square root of23. The square root of23 does not result in a simple integer or a fraction of two integers, and its decimal representation does not terminate or repeat.
Therefore, $\sqrt{23}$ is irrational.
Next, let's evaluate $-4.232323 \ldots$ .

STEP 6

The number $-4.232323 \ldots$ is a repeating decimal, which can be expressed as a ratio of two integers.
Therefore, $-4.232323 \ldots$ is not irrational.
Next, let's evaluate $0.151551555 \ldots$ .

STEP 7

The number $0.151551555 \ldots$ is a repeating decimal, which can be expressed as a ratio of two integers.
Therefore, $0.151551555 \ldots$ is not irrational.
Next, let's evaluate $0.3 \overline{5}$ .

STEP 8

The number $0.3 \overline{5}$ is a repeating decimal, which can be expressed as a ratio of two integers.
Therefore, $0.3 \overline{5}$ is not irrational.
Finally, let's evaluate $\pi$ .

STEP 9

The number $\pi$ is a well-known irrational number. It cannot be expressed as a ratio of two integers, and its decimal representation neither terminates nor repeats.
Therefore, $\pi$ is irrational.
In conclusion, the irrational numbers among the given options are $\sqrt{23}$ and $\pi$ .

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QuestionIdentify the irrational numbers from the list: $\sqrt{36}$ , $\sqrt{23}$ , $-4.232323 \ldots$ , $0.151551555 \ldots$ , $0.3 \overline{5}$ , $\pi$ .

Studdy Solution

STEP 1

STEP 2

Evaluate each number to determine if it is irrational.
First, let's evaluate $\sqrt{36}$ .

STEP 3

Calculate the square root of36.
$\sqrt{36} =6$

STEP 4

Since6 can be expressed as a ratio of two integers (6/1), $\sqrt{36}$ is not irrational.
Next, let's evaluate $\sqrt{23}$ .

STEP 5

Calculate the square root of23. The square root of23 does not result in a simple integer or a fraction of two integers, and its decimal representation does not terminate or repeat.
Therefore, $\sqrt{23}$ is irrational.
Next, let's evaluate $-4.232323 \ldots$ .

STEP 6

The number $-4.232323 \ldots$ is a repeating decimal, which can be expressed as a ratio of two integers.
Therefore, $-4.232323 \ldots$ is not irrational.
Next, let's evaluate $0.151551555 \ldots$ .

STEP 7

The number $0.151551555 \ldots$ is a repeating decimal, which can be expressed as a ratio of two integers.
Therefore, $0.151551555 \ldots$ is not irrational.
Next, let's evaluate $0.3 \overline{5}$ .

STEP 8

The number $0.3 \overline{5}$ is a repeating decimal, which can be expressed as a ratio of two integers.
Therefore, $0.3 \overline{5}$ is not irrational.
Finally, let's evaluate $\pi$ .

STEP 9

The number $\pi$ is a well-known irrational number. It cannot be expressed as a ratio of two integers, and its decimal representation neither terminates nor repeats.
Therefore, $\pi$ is irrational.
In conclusion, the irrational numbers among the given options are $\sqrt{23}$ and $\pi$ .

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QuestionIdentify the irrational numbers from the list: 36\sqrt{36}36​, 23\sqrt{23}23​, −4.232323…-4.232323 \ldots−4.232323…, 0.151551555…0.151551555 \ldots0.151551555…, 0.35‾0.3 \overline{5}0.35, π\piπ.

Studdy Solution

STEP 1

STEP 2

Evaluate each number to determine if it is irrational.First, let's evaluate 36\sqrt{36}36​.

STEP 3

Calculate the square root of36.36=6\sqrt{36} =636​=6

STEP 4

Since6 can be expressed as a ratio of two integers (6/1), 36\sqrt{36}36​ is not irrational.Next, let's evaluate 23\sqrt{23}23​.

STEP 5

Calculate the square root of23. The square root of23 does not result in a simple integer or a fraction of two integers, and its decimal representation does not terminate or repeat.Therefore, 23\sqrt{23}23​ is irrational.Next, let's evaluate −4.232323…-4.232323 \ldots−4.232323….

STEP 6

The number −4.232323…-4.232323 \ldots−4.232323… is a repeating decimal, which can be expressed as a ratio of two integers.Therefore, −4.232323…-4.232323 \ldots−4.232323… is not irrational.Next, let's evaluate 0.151551555…0.151551555 \ldots0.151551555….

STEP 7

The number 0.151551555…0.151551555 \ldots0.151551555… is a repeating decimal, which can be expressed as a ratio of two integers.Therefore, 0.151551555…0.151551555 \ldots0.151551555… is not irrational.Next, let's evaluate 0.35‾0.3 \overline{5}0.35.

STEP 8

The number 0.35‾0.3 \overline{5}0.35 is a repeating decimal, which can be expressed as a ratio of two integers.Therefore, 0.35‾0.3 \overline{5}0.35 is not irrational.Finally, let's evaluate π\piπ.

STEP 9

The number π\piπ is a well-known irrational number. It cannot be expressed as a ratio of two integers, and its decimal representation neither terminates nor repeats.Therefore, π\piπ is irrational.In conclusion, the irrational numbers among the given options are 23\sqrt{23}23​ and π\piπ.

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QuestionIdentify the irrational numbers from the list: $\sqrt{36}$ , $\sqrt{23}$ , $-4.232323 \ldots$ , $0.151551555 \ldots$ , $0.3 \overline{5}$ , $\pi$ .

Evaluate each number to determine if it is irrational.
First, let's evaluate $\sqrt{36}$ .

Calculate the square root of36.
$\sqrt{36} =6$

Since6 can be expressed as a ratio of two integers (6/1), $\sqrt{36}$ is not irrational.
Next, let's evaluate $\sqrt{23}$ .

Calculate the square root of23. The square root of23 does not result in a simple integer or a fraction of two integers, and its decimal representation does not terminate or repeat.
Therefore, $\sqrt{23}$ is irrational.
Next, let's evaluate $-4.232323 \ldots$ .

The number $-4.232323 \ldots$ is a repeating decimal, which can be expressed as a ratio of two integers.
Therefore, $-4.232323 \ldots$ is not irrational.
Next, let's evaluate $0.151551555 \ldots$ .

The number $0.151551555 \ldots$ is a repeating decimal, which can be expressed as a ratio of two integers.
Therefore, $0.151551555 \ldots$ is not irrational.
Next, let's evaluate $0.3 \overline{5}$ .

The number $0.3 \overline{5}$ is a repeating decimal, which can be expressed as a ratio of two integers.
Therefore, $0.3 \overline{5}$ is not irrational.
Finally, let's evaluate $\pi$ .

The number $\pi$ is a well-known irrational number. It cannot be expressed as a ratio of two integers, and its decimal representation neither terminates nor repeats.
Therefore, $\pi$ is irrational.
In conclusion, the irrational numbers among the given options are $\sqrt{23}$ and $\pi$ .