Solved on Feb 09, 2024

Find the second number that, when multiplied with $\frac{7}{3}$ , results in an irrational product. Options: $\frac{3}{7}$ , $2 \pi$ , 1, $\sqrt{3}$ .

STEP 1

Assumptions
1. The first number is $\frac{7}{3}$ .
2. The resulting product is an irrational number.
3. We need to determine which of the given numbers, when multiplied by $\frac{7}{3}$ , will result in an irrational number.

STEP 2

Recall the property of rational and irrational numbers: the product of a rational number and an irrational number is irrational.

STEP 3

Check each of the given numbers to see if it is rational or irrational.

STEP 4

The number $\frac{3}{7}$ is rational because it can be expressed as a ratio of two integers.

STEP 5

The number $2\pi$ is irrational because $\pi$ is a well-known irrational number, and multiplying it by a rational number (2) does not change its irrationality.

STEP 6

The number 1 is rational because it can be expressed as a ratio of two integers (for example, $\frac{1}{1}$ ).

STEP 7

The number $\sqrt{3}$ is irrational because it cannot be expressed as a ratio of two integers.

STEP 8

Multiply the first number $\frac{7}{3}$ by each of the given numbers to determine which product is irrational.

STEP 9

Multiply $\frac{7}{3}$ by $\frac{3}{7}$ :
$\frac{7}{3} \times \frac{3}{7} = \frac{7 \times 3}{3 \times 7} = \frac{21}{21} = 1$

STEP 10

Since the product in STEP_9 is 1, which is a rational number, $\frac{3}{7}$ is not a possible second number to result in an irrational product.

STEP 11

Multiply $\frac{7}{3}$ by $2\pi$ :
$\frac{7}{3} \times 2\pi = \frac{7 \times 2\pi}{3} = \frac{14\pi}{3}$

STEP 12

Since $\pi$ is irrational, the product in STEP_11 is also irrational. Therefore, $2\pi$ is a possible second number to result in an irrational product.

STEP 13

Multiply $\frac{7}{3}$ by 1:
$\frac{7}{3} \times 1 = \frac{7}{3}$

STEP 14

Since the product in STEP_13 is a rational number, 1 is not a possible second number to result in an irrational product.

STEP 15

Multiply $\frac{7}{3}$ by $\sqrt{3}$ :
$\frac{7}{3} \times \sqrt{3} = \frac{7\sqrt{3}}{3}$

STEP 16

Since $\sqrt{3}$ is irrational, the product in STEP_15 is also irrational. Therefore, $\sqrt{3}$ is a possible second number to result in an irrational product.

STEP 17

From the calculations, the numbers that, when multiplied by $\frac{7}{3}$ , result in an irrational number are $2\pi$ and $\sqrt{3}$ .
The possible second numbers are $2\pi$ and $\sqrt{3}$ .

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Find the second number that, when multiplied with $\frac{7}{3}$ , results in an irrational product. Options: $\frac{3}{7}$ , $2 \pi$ , 1, $\sqrt{3}$ .

STEP 1

Assumptions
1. The first number is $\frac{7}{3}$ .
2. The resulting product is an irrational number.
3. We need to determine which of the given numbers, when multiplied by $\frac{7}{3}$ , will result in an irrational number.

STEP 2

Recall the property of rational and irrational numbers: the product of a rational number and an irrational number is irrational.

STEP 3

Check each of the given numbers to see if it is rational or irrational.

STEP 4

The number $\frac{3}{7}$ is rational because it can be expressed as a ratio of two integers.

STEP 5

The number $2\pi$ is irrational because $\pi$ is a well-known irrational number, and multiplying it by a rational number (2) does not change its irrationality.

STEP 6

The number 1 is rational because it can be expressed as a ratio of two integers (for example, $\frac{1}{1}$ ).

STEP 7

The number $\sqrt{3}$ is irrational because it cannot be expressed as a ratio of two integers.

STEP 8

Multiply the first number $\frac{7}{3}$ by each of the given numbers to determine which product is irrational.

STEP 9

Multiply $\frac{7}{3}$ by $\frac{3}{7}$ :
$\frac{7}{3} \times \frac{3}{7} = \frac{7 \times 3}{3 \times 7} = \frac{21}{21} = 1$

STEP 10

Since the product in STEP_9 is 1, which is a rational number, $\frac{3}{7}$ is not a possible second number to result in an irrational product.

STEP 11

Multiply $\frac{7}{3}$ by $2\pi$ :
$\frac{7}{3} \times 2\pi = \frac{7 \times 2\pi}{3} = \frac{14\pi}{3}$

STEP 12

Since $\pi$ is irrational, the product in STEP_11 is also irrational. Therefore, $2\pi$ is a possible second number to result in an irrational product.

STEP 13

Multiply $\frac{7}{3}$ by 1:
$\frac{7}{3} \times 1 = \frac{7}{3}$

STEP 14

Since the product in STEP_13 is a rational number, 1 is not a possible second number to result in an irrational product.

STEP 15

Multiply $\frac{7}{3}$ by $\sqrt{3}$ :
$\frac{7}{3} \times \sqrt{3} = \frac{7\sqrt{3}}{3}$

STEP 16

Since $\sqrt{3}$ is irrational, the product in STEP_15 is also irrational. Therefore, $\sqrt{3}$ is a possible second number to result in an irrational product.

STEP 17

From the calculations, the numbers that, when multiplied by $\frac{7}{3}$ , result in an irrational number are $2\pi$ and $\sqrt{3}$ .
The possible second numbers are $2\pi$ and $\sqrt{3}$ .

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Find the second number that, when multiplied with 73\frac{7}{3}37​, results in an irrational product. Options: 37\frac{3}{7}73​, 2π2 \pi2π, 1, 3\sqrt{3}3​.

STEP 1

Assumptions1. The first number is 73\frac{7}{3}37​.2. The resulting product is an irrational number.3. We need to determine which of the given numbers, when multiplied by 73\frac{7}{3}37​, will result in an irrational number.

STEP 2

Recall the property of rational and irrational numbers: the product of a rational number and an irrational number is irrational.

STEP 3

Check each of the given numbers to see if it is rational or irrational.

STEP 4

The number 37\frac{3}{7}73​ is rational because it can be expressed as a ratio of two integers.

STEP 5

The number 2π2\pi2π is irrational because π\piπ is a well-known irrational number, and multiplying it by a rational number (2) does not change its irrationality.

STEP 6

The number 1 is rational because it can be expressed as a ratio of two integers (for example, 11\frac{1}{1}11​).

STEP 7

The number 3\sqrt{3}3​ is irrational because it cannot be expressed as a ratio of two integers.

STEP 8

Multiply the first number 73\frac{7}{3}37​ by each of the given numbers to determine which product is irrational.

STEP 9

Multiply 73\frac{7}{3}37​ by 37\frac{3}{7}73​:73×37=7×33×7=2121=1\frac{7}{3} \times \frac{3}{7} = \frac{7 \times 3}{3 \times 7} = \frac{21}{21} = 137​×73​=3×77×3​=2121​=1

STEP 10

Since the product in STEP_9 is 1, which is a rational number, 37\frac{3}{7}73​ is not a possible second number to result in an irrational product.

STEP 11

Multiply 73\frac{7}{3}37​ by 2π2\pi2π:73×2π=7×2π3=14π3\frac{7}{3} \times 2\pi = \frac{7 \times 2\pi}{3} = \frac{14\pi}{3}37​×2π=37×2π​=314π​

STEP 12

Since π\piπ is irrational, the product in STEP_11 is also irrational. Therefore, 2π2\pi2π is a possible second number to result in an irrational product.

STEP 13

Multiply 73\frac{7}{3}37​ by 1:73×1=73\frac{7}{3} \times 1 = \frac{7}{3}37​×1=37​

STEP 14

Since the product in STEP_13 is a rational number, 1 is not a possible second number to result in an irrational product.

STEP 15

Multiply 73\frac{7}{3}37​ by 3\sqrt{3}3​:73×3=733\frac{7}{3} \times \sqrt{3} = \frac{7\sqrt{3}}{3}37​×3​=373​​

STEP 16

Since 3\sqrt{3}3​ is irrational, the product in STEP_15 is also irrational. Therefore, 3\sqrt{3}3​ is a possible second number to result in an irrational product.

STEP 17

From the calculations, the numbers that, when multiplied by 73\frac{7}{3}37​, result in an irrational number are 2π2\pi2π and 3\sqrt{3}3​.The possible second numbers are 2π2\pi2π and 3\sqrt{3}3​.

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Find the second number that, when multiplied with $\frac{7}{3}$ , results in an irrational product. Options: $\frac{3}{7}$ , $2 \pi$ , 1, $\sqrt{3}$ .

Assumptions
1. The first number is $\frac{7}{3}$ .
2. The resulting product is an irrational number.
3. We need to determine which of the given numbers, when multiplied by $\frac{7}{3}$ , will result in an irrational number.

The number $\frac{3}{7}$ is rational because it can be expressed as a ratio of two integers.

The number $2\pi$ is irrational because $\pi$ is a well-known irrational number, and multiplying it by a rational number (2) does not change its irrationality.

The number 1 is rational because it can be expressed as a ratio of two integers (for example, $\frac{1}{1}$ ).

The number $\sqrt{3}$ is irrational because it cannot be expressed as a ratio of two integers.

Multiply the first number $\frac{7}{3}$ by each of the given numbers to determine which product is irrational.

Multiply $\frac{7}{3}$ by $\frac{3}{7}$ :
$\frac{7}{3} \times \frac{3}{7} = \frac{7 \times 3}{3 \times 7} = \frac{21}{21} = 1$

Since the product in STEP_9 is 1, which is a rational number, $\frac{3}{7}$ is not a possible second number to result in an irrational product.

Multiply $\frac{7}{3}$ by $2\pi$ :
$\frac{7}{3} \times 2\pi = \frac{7 \times 2\pi}{3} = \frac{14\pi}{3}$

Since $\pi$ is irrational, the product in STEP_11 is also irrational. Therefore, $2\pi$ is a possible second number to result in an irrational product.

Multiply $\frac{7}{3}$ by 1:
$\frac{7}{3} \times 1 = \frac{7}{3}$

Multiply $\frac{7}{3}$ by $\sqrt{3}$ :
$\frac{7}{3} \times \sqrt{3} = \frac{7\sqrt{3}}{3}$

Since $\sqrt{3}$ is irrational, the product in STEP_15 is also irrational. Therefore, $\sqrt{3}$ is a possible second number to result in an irrational product.

From the calculations, the numbers that, when multiplied by $\frac{7}{3}$ , result in an irrational number are $2\pi$ and $\sqrt{3}$ .
The possible second numbers are $2\pi$ and $\sqrt{3}$ .