Let x and y be non-zero, rational numbers and let z be an irrational number. In the first column of the table below is an expression. In the second column, determine whether the value of each expression is rational or irrational.
Expression Value of Expression
x + y
Rational
y + z
Rational
xy
Irrational
xz
Irrational
If x and y are rational numbers and non-zero, their sum x + y will also be a rational number. Therefore, the value of the expression x + y is rational.
If y is rational and z is irrational, their sum y + z will be irrational. Therefore, the value of the expression y + z is irrational.
When you multiply two non-zero rational numbers, the product will always be rational. Therefore, the value of the expression xy is rational.
When you multiply a rational number by an irrational number, the product will always be irrational. Therefore, the value of the expression xz is irrational.
To determine whether the given expressions are rational or irrational, we need to understand the properties of rational and irrational numbers.
Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot be written in such a form, and their decimal representations are non-repeating and non-terminating.
Let's analyze each expression:
1. x + y:
Since x and y are given as non-zero rational numbers, their sum will also be a rational number. Therefore, the value of this expression is rational.
2. y + z:
Here, y is a non-zero rational number, and z is given as an irrational number. The sum of a rational number and an irrational number is always an irrational number. Therefore, the value of this expression is irrational.
3. xy:
When we multiply two non-zero rational numbers, the result is always a rational number. Therefore, the value of this expression is rational.
4. xz:
In this case, x is a non-zero rational number, and z is given as an irrational number. The product of a rational number and an irrational number is always an irrational number. Therefore, the value of this expression is irrational.
In summary, the value of the expressions x + y and xy is rational, while the value of the expressions y + z and xz is irrational.
To determine whether the value of each expression is rational or irrational, we need to understand the properties of rational and irrational numbers.
A rational number can be expressed as a quotient of two integers (where the denominator is not zero). This means that a rational number can be written in the form of p/q, where p and q are integers.
An irrational number, on the other hand, cannot be expressed as a quotient of two integers. Irrational numbers are numbers that cannot be represented as terminating or repeating decimals and often involve square roots or other non-repeating patterns.
Now, let's analyze each expression in the table:
1. x + y:
The sum of two rational numbers is always a rational number. Since x and y are both rational numbers, their sum x + y will also be a rational number.
2. y + z:
Here, y is a rational number and z is an irrational number. When adding a rational number and an irrational number, the sum will be irrational. Therefore, y + z will be an irrational number.
3. xy:
When you multiply two rational numbers, the result is also a rational number. Since x and y are both rational numbers, their product xy will be a rational number.
4. xz:
Multiplying a rational number (x) by an irrational number (z) always results in an irrational number. Therefore, xz will be an irrational number.
To summarize:
- x + y is rational.
- y + z is irrational.
- xy is rational.
- xz is irrational.
By understanding the properties of rational and irrational numbers, we can determine the nature of the expressions given in the table.