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Apply the Closure Property to predict the answer. Will the sum of √25 and 1/8 be an irrational or rational number?(1 point)

Responses

It could be either, because there is one of each type of number.
It could be either, because there is one of each type of number.

The sum does not exist because a radical cannot be added to a fraction.
The sum does not exist because a radical cannot be added to a fraction.

Rational, because both addends are rational.
Rational, because both addends are rational.

Irrational, because one of the addends has a radical.

11 answers

To solve the problem, let's break it down:

  1. Calculate √25: \(\sqrt{25} = 5\), which is a rational number.
  2. Identify the other addend: \(1/8\) is also a rational number.

Now we have both addends:

  • \(5\) (rational)
  • \(1/8\) (rational)

According to the closure property of rational numbers, the sum of two rational numbers is also a rational number.

Now add them together: \[ 5 + \frac{1}{8} = \frac{40}{8} + \frac{1}{8} = \frac{41}{8} \]

Since \(\frac{41}{8}\) is a rational number, the correct response would be:

Rational, because both addends are rational.

When proving that the sum of two rational numbers is rational in the expression a+c devided by bd
, which value could be used for a in order to apply the proof?(1 point)
Responses

π
pi

√3

3/0

3
3

page 4 of 4

In the context of proving that the sum of two rational numbers is rational, we should consider the definitions of rational numbers:

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. This means any integer (like 3), or a fraction where both the numerator and denominator are integers (like \( \frac{1}{2} \)) is rational.

Looking at the provided options:

  • \( \pi \): This is an irrational number.
  • \( \sqrt{3} \): This is also an irrational number.
  • \( \frac{3}{0} \): This is undefined (you cannot divide by zero).
  • \( 3 \): This is a rational number.

In this case, the value that could be used for \( a \) in the proof is:

3 (as it is a rational number).

The product of √100
and 4/15
will be what type of number?(1 point)
Responses

irrational
irrational

rational
rational

zero
zero

an integer

Let's analyze the problem step by step:

  1. Calculate \( \sqrt{100} \): \[ \sqrt{100} = 10 \] which is a rational number (specifically, it is also an integer).

  2. Identify the other factor: \[ \frac{4}{15} \] which is also a rational number.

Now we find the product: \[ 10 \times \frac{4}{15} = \frac{10 \times 4}{15} = \frac{40}{15} \] This fraction can be simplified: \[ \frac{40}{15} = \frac{8}{3} \] which is still a rational number.

Since both factors (\( \sqrt{100} \) and \( \frac{4}{15} \)) are rational, their product will also be rational.

Therefore, the correct response is:

rational.

A rational number and an irrational number are multiplied together. The product is a rational number. What type of rational number must the product be?(1 point)
Responses

zero
zero

an integer
an integer

greater than 1
greater than 1

a perfect square

When you multiply a rational number by an irrational number, the product is always an irrational number, except in the case where the rational number is zero. In that scenario, the product would be zero.

To summarize:

  • If the rational number is zero, then the product (which is \(0 \times \) any number) will be zero, which is rational.
  • If the rational number is any non-zero rational number, the product will be irrational.

Thus, the only case when the product of a rational number and an irrational number results in a rational number is when the rational number is zero.

Therefore, the correct answer is:

zero.

In order to prove that the sum of two rational numbers, a and b, is rational, put the following steps in the correct order.

Step 1. Identify the sum as rational.

Step 2. Find a common denominator.

Step 3. Write a and b as fractions.

Step 4. Add the fractions.

(1 point)
Responses

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

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

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

Step 2, Step 3, Step 4, Step 1

To prove that the sum of two rational numbers \( a \) and \( b \) is rational, the logical sequence of steps should be followed as follows:

  1. Write \( a \) and \( b \) as fractions (Step 3) because rational numbers can be expressed in fraction form.

  2. Find a common denominator (Step 2) to facilitate the addition of the two fractions.

  3. Add the fractions (Step 4) to get the sum.

  4. Identify the sum as rational (Step 1) by noting that since both addends were rational and the sum of rational numbers is rational, the result is also rational.

Thus, the correct order of steps would be:

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

What type of number will result from 3.4059...−8/5
?(1 point)
Responses

Irrational, because one number is irrational.
Irrational, because one number is irrational.

Irrational, because both numbers are irrational.
Irrational, because both numbers are irrational.

Rational, because both numbers are rational.
Rational, because both numbers are rational.

Rational, because one number is rational.

Let's analyze the numbers given in the expression \( 3.4059... - \frac{8}{5} \):

  1. Identify \( 3.4059... \): This is a repeating decimal (if we assume the "..." indicates that the number continues indefinitely). Since it can be expressed as a fraction (a property of repeating decimals), it is a rational number.

  2. Identify \( \frac{8}{5} \): This is also a rational number, as it is the quotient of two integers.

Now we are subtracting two rational numbers:

\[ \text{Rational} - \text{Rational} = \text{Rational} \]

Since both \( 3.4059... \) and \( \frac{8}{5} \) are rational numbers, their difference will also be a rational number.

Therefore, the correct answer is:

Rational, because both numbers are rational.