Which of the following results in a sum or a product that is rational? Choose all that apply
(2 points)
4+9
pi * 5
3 overline 45 * 1.2
sqrt(16) * sqrt(4)
sqrt(7) * sqrt(2)
- 4 + 9 = 13 (rational sum)
- π * 5 = 5π (irrational product)
- 3.45 * 1.2 = 4.14 (rational product)
- sqrt(16) * sqrt(4) = 4 * 2 = 8 (rational product)
- sqrt(7) * sqrt(2) = sqrt(14) (irrational product)
The expressions 4 + 9, 3.45 * 1.2, and sqrt(16) * sqrt(4) result in a sum or a product that is rational.
To determine which of the given expressions results in a sum or product that is rational, we need to evaluate each expression.
1) 4 + 9 = 13 -> This is a sum and it is a rational number.
2) π * 5 -> The product of a rational number (5) and an irrational number (π) is irrational.
3) 3 overline 45 * 1.2 -> The first number is a recurring decimal, but when multiplied by 1.2 (a rational number), the result is rational.
4) sqrt(16) * sqrt(4) -> sqrt(16) = 4 and sqrt(4) = 2, so the product is 4 * 2 = 8, which is rational.
5) sqrt(7) * sqrt(2) -> The product of two irrational numbers (sqrt(7) and sqrt(2)) is irrational.
Therefore, the expressions that result in a sum or product that is rational are:
4 + 9
3 overline 45 * 1.2
sqrt(16) * sqrt(4)
To determine which of the given expressions result in a sum or product that is rational, we need to understand what rational numbers are and how they are obtained.
Rational numbers are those that can be expressed as a ratio (fraction) of two integers, where the denominator is not zero. In other words, rational numbers can be written in the form a/b, where a and b are integers.
Now, let's analyze each expression:
1. 4 + 9:
In this expression, we are adding the integers 4 and 9. The sum of two integers is always an integer, and integers are a subset of rational numbers. Therefore, the sum 4 + 9 is rational.
2. π * 5:
Here, we are multiplying the irrational number π (pi) by the integer 5. When any real number is multiplied by an integer, the product is also a real number. Since π is an irrational number and real numbers encompass rational numbers, the product π * 5 is a real number, including rational numbers.
3. 3 overline 45 * 1.2:
The expression 3 overline 45 represents a repeating decimal, also known as a rational number. Multiplying a rational number by a real number (in this case, 1.2) will result in another rational number.
4. √16 * √4:
The expression √16 represents the square root of 16, which equals 4, and √4 represents the square root of 4, which equals 2. Since both 4 and 2 are integers, the product of two integers is always an integer and, therefore, a rational number.
5. √7 * √2:
In this expression, we have the square root of 7 multiplied by the square root of 2. Both √7 and √2 are irrational numbers because they cannot be expressed as a fraction of two integers. Therefore, the product of these two irrational numbers is also an irrational number.
Based on the explanations above, the expressions that result in a sum or product that is rational are:
- 4 + 9
- π * 5
- 3 overline 45 * 1.2
- √16 * √4
Therefore, the correct choices are:
- 4 + 9
- π * 5
- 3 overline 45 * 1.2
- √16 * √4