Understanding Rationalizing in Mathematics
Discover the essential concept of rationalizing in mathematics, focusing on not leaving radicals in the denominator of a fraction. Learn how to change the denominator of a fraction without altering its value by multiplying the numerator and denominator by the same number. Explore the process of rationalizing to convert the denominator into a rational number, illustrated step-by-step with examples.
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Presentation Transcript
Rationalizing 3 6 6 2
There is an agreement in mathematics that we don t leave a radical in the denominator of a fraction. 1 3
So how do we change the denominator of a fraction? (Without changing the value of the fraction, of course.) 1 3
The same way we change the denominator of any fraction! (Without changing the value of the fraction, of course.) 1 3 3 1 = = 4 3 12 4
We multiply the denominator and the numerator by the same number. 1 3 3 1 = = 4 3 12 4
By what number can we multiply 3 to change it to a rational number? 1 3
The answer is . . . . . . by itself! 2 3 ( ) = = 3 3 3 1 3
Remember, ( )2 = n n n is the number we square to get n. So when we square it, we d better get n.
In our fraction, to get the radical out of the denominator, we can multiply numerator and denominator by . 3 1 3
In our fraction, to get the radical out of the denominator, we can multiply numerator and denominator by . 3 1 1 3 3 = = 3 3 3 3
Because we are changing the denominator to a rational number, we call this process rationalizing. 3 1 = 3 3
Rationalize the denominator: 2 4 2 4 2 4 = = = 2 2 2 2 2 2
Rationalize the denominator: 8 96 8 12 = = = 12 12 12 12 6 4 6 = 3 12 3
When there is a binomial with a radical in the denominator of a fraction, you find the conjugate and multiply. This gives a rational denominator. Ex: 5 +6 Conjugate: 5 6 3 2 2 Conjugate: 3+2 2 What is conjugate of 2 7+3? Answer: 2 7 3
5 + 6 5 3 Simplify: 5 +6 5 3= 5 +6 5 3 5 +3 5 +3 Multiply by the conjugate. 5+3 5 +6 5 +18 5 9 FOIL numerator and denominator. Next
2 Simplify 2 5 2 + 5 + 2 5 2 5 2 2(5) 2 2 2 5 ( 2) 2 10 2 2 23 10 2 2 25 2 =
23 +9 5 4 Combine like terms Try this on your own: 6 Answer: 3 6 2 3 3+ 2 7
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