Question

Rationalizing factor of 5⅓+5-⅓​

Answer 1

Answer:

How do I rationalize 5^1/3 - 5^-1/3 with its rationalizing factor?

Your question is how to rationalize the above expression.

The above answer gives you the shortest way to rationalize the expression.

For finding rationalizing factor you can use this method. This is somewhat complex .

Take 5^1/3=a & 5^-1/3=b

Now find the lcm if the exponents.

In this case since the denominator is same therefore the LCM is 3

Now a^3=5 & b^3=5^-1

Since a and b are rational therefore a^3-b^3 is also rational

a^3-b^3=(a-b)(a^2+ab+b^2)

Rationalizing factor of a-b=(5^1/3–5^-1/3)

=5^2/3+5^–2/3+5^2/3*5^-2/3

Now just solve the above expression and get the rationalizing factor

Step-by-step explanation:

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Answer 2

Given,

$5^{\frac{1}{3} } +5^{\frac{1}{3} }$

To Find,

Rationalizing factor of $5^{\frac{1}{3} } 5^{\frac{1}{3} }$

Solution,

from formula

$a^{3} +b^{3}=(a+b)(a^{2}-ab +b^{2} )$

The rationalizing factor of (a+b) is ($a^{2}-ab+ b^{2}$)

therefore,

Rationalizing factor  $5^{\frac{1}{3} } +5^{\frac{1}{3} }$ is $5^{2(^{\frac{1}{3} } )} -5^{\frac{1}{3} } .5^{-\frac{1}{3} } +5^{2(\frac{1}{3}) }$

    =>$5^{(^{\frac{2}{3} } )} -5^{\frac{1}{3} } .5^{-\frac{1}{3} } +5^{\frac{2}{3} }$

    =>$5^{(^{\frac{2}{3} } )} -5^{\frac{1}{3}- \frac{1}{3} +5^{\frac{2}{3} }$

    =>$5^{2(^{\frac{1}{3} } )} -1+5^{\frac{1}{3} }$

Hence the Rationalizing factor will be $5^{2(^{\frac{1}{3} } )} -1+5^{\frac{1}{3} }$

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