Lesson 3
1. Lesson 3
1.7. Explore 3
Module 5: Radicals
Introducing Conjugates
The binomial factors for the difference of squares are called conjugates. In the equation 
, the conjugates are (x − y) and (x + y). Conjugates may contain radicals; for example, 
.
Example
If you had difficulty solving in Try This 3, this example should help. You may want to revisit Try This 3 after reading this example.
This example shows how to rationalize an expression with a binomial denominator containing a radical. First, determine the conjugate of the denominator. Then multiply both the numerator and the denominator by this conjugate. This will eliminate the radical from the denominator.
Consider 
.
The binomial denominator is 
.
The conjugate of 
. Multiply the numerator and denominator by the conjugate 
.
Notice that by multiplying the expression by the conjugate, the radical in the numerator is simplified to an integer. This will be the case every time.
Simplify:
More Examples
“How to Rationalize the Denominator” summarizes the different strategies you have now learned to eliminate the radical from the denominator. The video progresses from basic examples to more complex problems.
Self-Check 2
- Simplify the following radicals by rationalizing the denominator. Verify your answers using a calculator.
-
The highest diving platforms at pools are 10 m above the water. The time a diver takes to hit the water can be calculated using the radical equation
. Assume the distance, d , is –10 m. The acceleration due to gravity, a, is –9.81 m/s2.
The negative sign in both the values given indicates that the direction is downward, since the diver is falling. Since you cannot take the square root of negative numbers, rationalize the denominator before substituting the values in for the variables. Use the resulting expression and the data given to find the time for a dive to the nearest tenth of a second. Answer
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