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Rationalizing the Denominator-Two terms

by Nick
(Surrey,B.C, Canada)











































_ _
2/3 + /7
--------
_ _
3/3 - /7



_ _ _ _
2/3 + /7 ( 3/3 + /7 )
= -------- . -------------
_ _ _ _
3/3 - /7 ( 3/3 + /7 )

_ _
18 + 2/21 + 3/21 + 7
= --------------------
20

_
5 + 5/21
= ---------
4

( I know my answer is wrong because the answer from my textbook is :
5 + 5/21
= --------- ) Please help.Thank you.
4

Comments for Rationalizing the Denominator-Two terms

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Apr 01, 2011
Rationalizing the Denominator-Two terms
by: Staff


The question:

by Nick
(Surrey, B.C, Canada)

_ _
2/3 + /7
--------
_ _
3/3 - /7



_ _ _ _
2/3 + /7 ( 3/3 + /7 )
= -------- . -------------
_ _ _ _
3/3 - /7 ( 3/3 + /7 )

_ _
18 + 2/21 + 3/21 + 7
= --------------------
20

_
5 + 5/21
= ---------
4

( I know my answer is wrong because the answer from my textbook is :
5 + 5/21
= --------- ) Please help.Thank you.
4


The answer:

Your approach to rationalize the denominator is correct.

I’m going to duplicate your steps below:

original expression


2√(3) + √(7)
----------------
3√(3) - √(7)

Multiply the numerator and denominator by 3√(3) + √(7)
(this is also the approach you took)

= {[2√(3) + √(7)]/[ 3√(3) - √(7)]}*{[3√(3) + √(7)]/[ 3√(3) + √(7)]}

= {[2√(3) + √(7)]*[3√(3) + √(7)]}/{[3√(3) - √(7)]*[ 3√(3) + √(7)]}

Up to this point, we are in agreement

= {[2√(3) + √(7)]*[3√(3) + √(7)]}/20


= [25 + 5√(21)]/20

= 5*[5 + √(21)]/20

= 5*[5 + √(21)]/20

= 1*[5 + √(21)]/4

= [5 + √(21)]/4


The final answer is: [5 + √(21)]/4


Let’s take one more step and verify the answer.

Using a calculator, calculate a decimal equivalent for both the original expression and the rationalized version. Compare the two answers to ensure they are the same.

You can use any calculator. I prefer Microsoft excel because the complete expression can be entered at one time.

Original expression:

= [2√(3) + √(7)]/[ 3√(3) - √(7)]

To use excel, enter the expression as follows:

= (2*sqrt(3) + sqrt(7))/(3*sqrt(3) - sqrt(7))

= 2.395644

Rationalized expression:

= [5 + √(21)]/4

To use excel, enter the expression as follows:

= (5 + sqrt(21))/4

= 2.395644


Both calculations match exactly, so the rationalized version is correct:

= [5 + √(21)]/4


Where did you go wrong?

It looks like you made a minor error in your arithmetic when you multiplied the following expression:

[2√(3) + √(7)]*[3√(3) + √(7)]

Your answer:

= 18 + 2√21 + 3√21 + 7 (this is correct)

≠ 5 + 5√21 (incorrect)

It should be

= 18 + 2√21 + 3√21 + 7 (this is correct)

= 18 + 7 + 2√21 + 3√21

= 25 + 5√21




Thanks for writing.


Staff
www.solving-math-problems.com



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