Evaluate the sums and determine which statement is true.

Question

Column A
Column B
0.7 + 1.8
1 + 0.25

Statements:
1. Only Column A has a rational sum.
2. Only Column B has a rational sum.
3. Both Column A and Column B have rational sums.
shorter explaination

Answer 1

To evaluate the sums:

Column A:
$0.7+1.8=2.5$ (rational)

Column B:
$1+0.25=1.25$ (rational)

Both sums are rational.

Thus, the correct statement is: 3. Both Column A and Column B have rational sums.

Answer 2

The product of a multiplication problem is 144. What do you know about the factors?

Answer 3

If the product of a multiplication problem is 144, then the factors are the numbers that can be multiplied together to get 144.

Here are some key points about the factors of 144:

1. Pairing Factors: Factors can be paired. For example, $1\times 144$, $2\times 72$, $3\times 48$, $4\times 36$, $6\times 24$, $8\times 18$, and $9\times 16$ are pairs of factors whose product is 144.

2. Total Factors: 144 has multiple factors. The complete list of factors includes: $1,2,3,4,6,8,9,12,16,18,24,36,48,72,144$.

3. Prime Factorization: The prime factorization of 144 is $2^4\times 3^2$. This means that the number 144 can be expressed as the product of prime factors raised to their respective powers.

4. Even and Odd Factors: All the factors of 144 are even except for 1 and 3.

5. Perfect Square: Since $12\times 12=144$, it is a perfect square.

Overall, factors of 144 are integers that can be multiplied together to give 144, and they can be both even and odd.

Answer 4

Arrange the steps to prove that the
sum t = r + s is irrational if r is rational and s is irrational.

1. Assume for contradiction that t is rational.
2. Write r and s in fractional form.
3. Find a common denominator.
4. Realize a contradiction.

Answer 5

To prove that the sum $t=r+s$ is irrational if $r$ is rational and $s$ is irrational, the steps can be arranged as follows:

1. Assume for contradiction that t is rational.
2. Write r and s in fractional form.
3. Find a common denominator.
4. Realize a contradiction.

This arrangement logically follows the process of proof by contradiction. You start by assuming the opposite of what you want to prove (that the sum $t$ is rational), express the rational and irrational numbers in a suitable form, manipulate them mathematically, and ultimately arrive at a contradiction that reveals the flaw in the initial assumption.

Answer 6

What type of number will the product of π and 5 be?

Answer 7

The product of $\pi$ (which is an irrational number) and $5$ (which is a rational number) will also be an irrational number.

In general, the product of a nonzero rational number and an irrational number is always irrational. Thus, $\pi \times 5$ is irrational.

Answer 8

Complete the expression adding the radicals.
⬜12 +227=14⬜

Answer 9

To complete the expression by adding the radicals, we first simplify each term.

1. Simplify $\sqrt{12}$: $$\sqrt{12}=\sqrt{4\cdot 3}=\sqrt{4}\cdot \sqrt{3}=2\sqrt{3}$$

2. Simplify $\sqrt{27}$: $$\sqrt{27}=\sqrt{9\cdot 3}=\sqrt{9}\cdot \sqrt{3}=3\sqrt{3}$$

Now we can add these two simplified radicals together: $$2\sqrt{3}+3\sqrt{3}=(2+3)\sqrt{3}=5\sqrt{3}$$

Now we can express it as follows: $$\boxed{{\displaystyle 5\sqrt{3}=14\sqrt{3}}}$$ So the completed expression should read: $$\sqrt{12}+\sqrt{27}=5\sqrt{3}$$

However, the original expression seems to indicate there's a specific value for the right side of the equation, which may be $14\sqrt{3}$ but in a true mathematical sense, the left side should only equal $5\sqrt{3}$ unless we're discussing a different context or comparing terms. If you would like clarification or to change the format, please let me know!