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Rationalize the Numerator of {∛(4b²)}/4










































Rationalize a Numerator containing a cube root

An irrational numerator can be can be changed into a rational number through a process called rationalizing the numerator.

How do I rationalize the numerator in the cubed root of {∛(4b²)}/4?

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Sep 04, 2012
Rationalize the Numerator
by: Staff


Answer:
Part I


[∛(4b²)] / 4

= [∛(4b²)] / 4


You can remove the radical from the numerator in your problem [∛(4b²)] by multiplying the numerator by itself two more times: [∛(4b²)]*[∛(4b²)]*[∛(4b²)] = 4b²


However, in order to preserve the value of the original fraction, both the numerator and denominator must each be multiplied by the same amount: [∛(4b²)].

To apply this concept, multiply the original fraction by [∛(4b²)] / [∛(4b²)] * [∛(4b²)] / [∛(4b²)] . The fraction [∛(4b²)] / [∛(4b²)] is equal to 1, so the original fraction is merely being multiplied by 1. As you can see by the following illustration, the value of the original fraction has not been changed.


= [original fraction]

= [original fraction] * [∛(4b²)] / [∛(4b²)] * [∛(4b²)] / [∛(4b²)]

= [original fraction] * 1 * 1

= [original fraction]


Therefore,

= [original fraction] * 1 * 1

= {[∛(4b²)] / 4} * 1 * 1

= {[∛(4b²)] / 4} * {∛(4b²) / ∛(4b²)} * {∛(4b²) / ∛(4b²)}

= [∛(4b²) / 4] * [∛(4b²) / ∛(4b²)] * [∛(4b²) / ∛(4b²)]


Multiply all three numerators and multiply all three denominators, just as you would when multiplying any three fractions:



= [∛(4b²) * ∛(4b²) * ∛(4b²)] / [4 * ∛(4b²) * ∛(4b²)]


It is sometimes easier to convert the radical signs to fractional exponents, and then work with the exponents. It is not necessary, but you may find it convenient.


= [(4b²) * (4b²) * (4b²)] / [4 * (4b²) * (4b²)]

Add the exponents as shown. As you can see, the fractional exponent (the cube root) in the numerator disappears.

= (4*b²)⅓+⅓+⅓ / [4 * (4b²)⅓+⅓]

= (4*b²)1 / [4 * (4b²)]

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Sep 04, 2012
Rationalize the Numerator
by: Staff


----------------------------------

Part II


The factor (4) in the numerator will cancel the (4) in the denominator.

= (4*b²) / [4 * (4b²)]

= (4 / 4) * [b² / (4b²)]

= (4 / 4) * [b² / (4b²)]

= (1) * [b² / (4b²)]

= b² / (4b²)

= b² / (2²b²)


When evaluating an exponent of an exponent, multiply the exponents.

= b² / (2(2*⅔)) * b(2*⅔))

= b² / (2(4/3) * b(4/3))

= b² / 21+⅓b1+⅓

= b² / (21 * 2 * b1 * b)

= b² / (2b * 2 * b)


The factor (b) in the numerator will cancel the (b) in the denominator.


= (b / b) * b / (2 * 2 * b)


= ( b / b ) * b / (2 * 2 * b)

= (1) * b / (2 * 2 * b)

= b / (2 * 2 * b)

= b / (2 * (2b)

= b / (2 * ∛(2b)

= b / (2∛(2b)




The final answer is:


b / [2∛(2b)]




Thanks for writing.

Staff
www.solving-math-problems.com



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