Adding, subtracting, multiplying, and dividing rational expressions is similar to performing the same operations on rational numbers. Using examples for each operation, support this statement.
To support the statement that adding, subtracting, multiplying, and dividing rational expressions is similar to performing the same operations on rational numbers, let's look at examples of each operation:
1. Addition:
Rational Numbers: Let's say we have the rational numbers 1/2 and 3/4. To add them, we need to find a common denominator, which in this case is 4. Thus, the addition would be (1/2) + (3/4) = (2/4) + (3/4) = 5/4.
Rational Expressions: Now, let's consider the rational expressions (x/2) and (3x/4). To add them, we also need to find a common denominator. Thus, the addition would be (x/2) + (3x/4) = (2x/4) + (3x/4) = (5x/4).
From this example, we can observe that adding rational expressions is similar to adding rational numbers, as we find a common denominator and then perform the addition.
2. Subtraction:
Rational Numbers: Let's consider the rational numbers 3/4 and 1/2. To subtract them, we still need a common denominator, which is 4. The subtraction would be (3/4) - (1/2) = (3/4) - (2/4) = 1/4.
Rational Expressions: Now, consider the rational expressions (2x/4) and (x/2). To subtract them, we similarly find a common denominator. The subtraction would be (2x/4) - (x/2) = (2x/4) - (2x/4) = 0.
Again, we can see that subtracting rational expressions follows the same procedure as subtracting rational numbers, by finding a common denominator and then performing the subtraction.
3. Multiplication:
Rational Numbers: Let's take the rational numbers 1/2 and 3/4. To multiply them, we just multiply the numerators and denominators. The multiplication would be (1/2) * (3/4) = 3/8.
Rational Expressions: Now, consider the rational expressions (x/2) and (3x/4). To multiply them, we also multiply the numerators and denominators. The multiplication would be (x/2) * (3x/4) = (3x^2/8).
Once again, multiplying rational expressions is similar to multiplying rational numbers, where we simply multiply the numerators and denominators.
4. Division:
Rational Numbers: Let's consider the rational numbers 1/2 and 3/4. To divide them, we multiply the first number by the reciprocal of the second number. The division would be (1/2) / (3/4) = (1/2) * (4/3) = 4/6 = 2/3.
Rational Expressions: Now, consider the rational expressions (x/2) and (3x/4). To divide them, we similarly multiply the first expression by the reciprocal of the second expression. The division would be (x/2) / (3x/4) = (x/2) * (4/3x) = 4/6 = 2/3.
Again, we can see that division of rational expressions follows the same procedure as division of rational numbers, by multiplying the first term by the reciprocal of the second term.
Therefore, adding, subtracting, multiplying, and dividing rational expressions follows the same principles as performing these operations on rational numbers. We find common denominators when necessary and perform the operation accordingly.