Math

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

Studdy Solution

STEP 1

Assumptions
1. The first number is 73\frac{7}{3}.
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}, 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} is rational because it can be expressed as a ratio of two integers.

STEP 5

The number 2π2\pi is irrational because π\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, 11\frac{1}{1}).

STEP 7

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

STEP 8

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

STEP 9

Multiply 73\frac{7}{3} by 37\frac{3}{7}:
73×37=7×33×7=2121=1\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, 37\frac{3}{7} is not a possible second number to result in an irrational product.

STEP 11

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

STEP 12

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

STEP 13

Multiply 73\frac{7}{3} by 1:
73×1=73\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 73\frac{7}{3} by 3\sqrt{3}:
73×3=733\frac{7}{3} \times \sqrt{3} = \frac{7\sqrt{3}}{3}

STEP 16

Since 3\sqrt{3} is irrational, the product in STEP_15 is also irrational. Therefore, 3\sqrt{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}, result in an irrational number are 2π2\pi and 3\sqrt{3}.
The possible second numbers are 2π2\pi and 3\sqrt{3}.

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