What is the process we follow when multiplying and dividing radical expressions? Explain the process and demonstrate with an example.
Multiplication of Rational Numbers:
1. Consider the factors of the numerator and of the denominator.
2. If there are factors common to the numerator and the denominator, reduce by dividing out the common factors.
3. Multiply all the numerators together and multiply all the denominators together.
Note that the steps 2 and 3 may be interchanged.
Dividing is the same thing as multiplying, you just flip the second fraction and multiply, you aren't actually dividing.
Example:
(x+1)/(x-2)(x+3) * (2)(x+3)/(x)(x+1)
The x+1's cancel out, as well as the x+3's. And you are left with 2 in the numerator and x(x-2) in the denominator.
When multiplying and dividing radical expressions, we follow certain steps to simplify the expressions. Let me explain the process step by step and demonstrate with an example.
To multiply radical expressions:
1. Multiply the numbers outside the radical sign (the coefficient) and simplify if possible.
2. Multiply the numbers inside the radical sign (the radicand) and simplify if possible.
3. Multiply the resulting coefficients and radicands separately.
4. Combine the multiplied coefficients and radicands into a single expression.
5. Simplify the expression if possible by performing any remaining calculations.
To divide radical expressions:
1. Divide the numbers outside the radical sign (the coefficient) and simplify if possible.
2. Divide the numbers inside the radical sign (the radicand) and simplify if possible.
3. Divide the resulting coefficients and radicands separately.
4. Combine the divided coefficients and radicands into a single expression.
5. Simplify the expression if possible by performing any remaining calculations.
Let's demonstrate with an example:
Example: Simplify (3√5)(4√3)
Step 1: Multiply the coefficients: 3 * 4 = 12
Step 2: Multiply the radicands: √5 * √3 = √15
Step 3: Multiply the coefficients and radicands separately: 12 * √15
Step 4: Combine the multiplied coefficients and radicands into a single expression: 12√15
Therefore, the simplified expression is 12√15.
Now let's demonstrate the division process with an example:
Example: Simplify (4√8) / (2√2)
Step 1: Divide the coefficients: 4 / 2 = 2
Step 2: Divide the radicands: √8 / √2 = √(8/2) = √4 = 2
Step 3: Divide the coefficients and radicands separately: 2 / 2
Step 4: Combine the divided coefficients and radicands into a single expression: 1
Therefore, the simplified expression is 1.
Remember, simplifying radical expressions involves multiplying or dividing the coefficients and radicands separately and then combining them into a single expression.