Evaluate the two sums and determine which statement is true. Enter the correct number associated with your response.

Question

Column A Column B
0.5+3/4


√3+2/5


Statement #1: Only Column A has a rational sum.

Statement #2: Only Column B has a rational sum.

Statement #3: Column A and Column B both have rational sums.

(1 point)
The simplified values of both columns indicate that Statement #
is true.

Answer 1

First, let's evaluate the sums in both columns:

Column A: \[ 0.5 + \frac{3}{4} \] We can convert \(0.5\) to a fraction: \[ 0.5 = \frac{1}{2} \] Now we can add \(\frac{1}{2} + \frac{3}{4}\). To add these fractions, we need a common denominator, which is \(4\): \[ \frac{1}{2} = \frac{2}{4} \] Now we can add: \[ \frac{2}{4} + \frac{3}{4} = \frac{5}{4} \] This is a rational number.

Column B: \[ \sqrt{3} + \frac{2}{5} \] The term \(\sqrt{3}\) is an irrational number, and \(\frac{2}{5}\) is a rational number. The sum of a rational and an irrational number is always irrational: \[ \sqrt{3} + \frac{2}{5} \text{ is irrational.} \]

Now we have determined the nature of the sums:

• Column A: \( \frac{5}{4} \) is rational.
• Column B: \( \sqrt{3} + \frac{2}{5} \) is irrational.

Now let's evaluate the statements:

• Statement #1: Only Column A has a rational sum. (True)
• Statement #2: Only Column B has a rational sum. (False)
• Statement #3: Column A and Column B both have rational sums. (False)

Thus, the correct statement is Statement #1.

The response is: 1.

Answer 2

The product of a multiplication problem is √225−−−
. What do you know about the factors?(1 point)
Responses

They are both rational because the product is rational.
They are both rational because the product is rational.

At least one factor is irrational because the product is irrational.
At least one factor is irrational because the product is irrational.

One factor is rational, but the other is irrational.
One factor is rational, but the other is irrational.

It cannot be predicted based on the information given.
It cannot be predicted based on the information given.

Answer 3

First, let’s simplify the product:

\[ \sqrt{225} = 15 \]

Since \(15\) is a rational number, we can analyze the possible factors:

1. Both factors can be rational: For example, \(1 \times 15 = 15\) or \(3 \times 5 = 15\).

2. One factor can be irrational, but the other must also be rational: An example would be \( \sqrt{9} \) (which is \(3\), a rational number) and \( \sqrt{25} \) (which is \(5\), also a rational number).

3. The factors can also be irrational: For instance, \( \sqrt{225} \) can also be expressed in terms of its factors:

   • \( (\sqrt{15})^2 = 15\) (Here, both \( \sqrt{15} \) and \( \sqrt{15} \) are irrational. However, their product is rational).

The conclusion is that if the product is rational, the factors can either both be rational, one factor can be rational and the other irrational such that they can multiply to give rational, or both could potentially be irrational but their product still results in a rational value.

From the options provided and based on the information that we have, the most fitting choice is:

They are both rational because the product is rational. (as there are rational factors that could result in the rational product).

So the correct response is:

They are both rational because the product is rational.

Answer 4

no explaining
Put the steps to the proof that the sum t=r+s
is irrational if r is a rational and s is rational in the correct sequence.

Step 1. Subtract, writing s as a fraction.

Step 2. For a contradiction, assume that t is rational, and write r and t as fractions.

Step 3. Realize a contradiction.

Step 4. Find a common denominator.

(1 point)
Responses

Step 3, Step 4, Step 2, Step 1
Step 3, Step 4, Step 2, Step 1

Step 2, Step 4, Step 1, Step 3
Step 2, Step 4, Step 1, Step 3

Step 4, Step 2, Step 1, Step 3
Step 4, Step 2, Step 1, Step 3

Step 1, Step 3, Step 2, Step 4

Answer 5

The correct sequence of steps to prove that the sum \( t = r + s \) is irrational if \( r \) is rational and \( s \) is rational is:

Step 2: For a contradiction, assume that \( t \) is rational, and write \( r \) and \( t \) as fractions.

Step 4: Find a common denominator.

Step 1: Subtract, writing \( s \) as a fraction.

Step 3: Realize a contradiction.

Therefore, the correct response is:

Step 2, Step 4, Step 1, Step 3.

Answer 6

What type of number will the product of π
and 3/4
be?(1 point)
Responses

a rational number
a rational number

a repeating decimal
a repeating decimal

an irrational number
an irrational number

a radical

Answer 7

The product of \( \pi \) (an irrational number) and \( \frac{3}{4} \) (a rational number) is:

\[ \pi \times \frac{3}{4} \]

The product of a rational number and an irrational number is always an irrational number.

Therefore, the correct response is:

an irrational number.

Answer 8

Complete the expression adding the radicals.(1 point)

_√5 + 3√45 = √13 _

Answer 9

To complete the expression by adding \( \sqrt{5} + 3\sqrt{45} \), we first simplify \( 3\sqrt{45} \).

1. Simplify \( \sqrt{45} \): \[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \]

2. Now substitute back into the expression: \[ 3\sqrt{45} = 3(3\sqrt{5}) = 9\sqrt{5} \]

3. Now add \( \sqrt{5} \) and \( 9\sqrt{5} \): \[ \sqrt{5} + 9\sqrt{5} = 10\sqrt{5} \]

Thus, the complete expression is:

\[ 10\sqrt{5} = \sqrt{13} \]

Therefore, we complete the expression as:

10√5 = √13.

Answer 10

13√_?