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Answer 1

Hello Dear Friend

Question :

$\bf{Rationalize\:\:numerator\:\:\dfrac{\sqrt{1+C^3}-\sqrt{1-C^3}}{C^2}}$

Given :

$\bf{\dfrac{\sqrt{1+C^3}-\sqrt{1-C^3}}{C^2}}$

Solution :

$\bf{Multiply\:by\:the\:conjugate}\:\dfrac{\sqrt{1+C^3}+\sqrt{1-C^3}}{\sqrt{1+C^3}+\sqrt{1-C^3}}$

✯ To rationlize the numerator, multiply the numerator and denominator

by conjugate of the radical : $\bf{\sqrt{1+C^{3}}-\sqrt{1-C^{3}}$

$=\tt{\dfrac{\left(\sqrt{1+C^3}-\sqrt{1-C^3}\right)\left(\sqrt{1+C^3}+\sqrt{1-C^3}\right)}{C^2\left(\sqrt{1+C^3}+\sqrt{1-C^3}\right)}}$

➜  Simplify the numerator

$\bf{\left(\sqrt{1+C^3}-\sqrt{1-C^3}\right)\left(\sqrt{1+C^3}+\sqrt{1-C^3}\right)}$

$\sf{Apply\:Difference\:of\:Two\:Squares\:Formula:\:}\left(a-b\right)\left(a+b\right)=a^2-b^2}$

$\sf{a=\sqrt{1+C^3},\:b=\sqrt{1-C^3}}$

$\sf{a^2-b^2=\left(\sqrt{1+C^3}\right)^2-\left(\sqrt{1-C^3}\right)^2}$

Simplify $\sf{\left(\sqrt{1+C^3}\right)^2-\left(\sqrt{1-C^3}\right)^2}$

$\sf{\left(\sqrt{1+C^3}\right)^2=1+C^3}\\\\\sf{\left(\sqrt{1-C^3}\right)^2=1-C^3}$

$\bf{=1+C^3-\left(-C^3+1\right)}$

$\sf{\left(1-C^3\right):\quad -1+C^3}$

$\sf{=1+C^3-1+C^3}$

➜ Simplify $\sf{1+C^3-1+C^3}$

$\mathrm{Group\:like\:terms}$

$=C^3+C^3+1-1$

$\mathrm{Add\:similar\:elements:}\:C^3+C^3=2C^3$

$=2C^3+1-1$

$=2C^3 + 0$

$=2C^3$

$=\bf{\dfrac{2C^3}{C^2\left(\sqrt{1+C^3}+\sqrt{1-C^3}\right)}}$

$\mathrm{Apply\:exponent\:rule}:\quad \dfrac{x^a}{x^b}=x^{a-b}$

$\dfrac{C^3}{C^2}=C^{3-2}$

$=\dfrac{2C^{3-2}}{\sqrt{C^3+1}+\sqrt{-C^3+1}}$

$\huge\black\boxed{\bf{\frac{\sqrt{1+C^3}-\sqrt{1-C^3}}{C^2}=\frac{2C}{\sqrt{C^3+1}+\sqrt{-C^3+1}}}}$

Hope you got the answer required !!!

Answer 2

Answer:

ationalizenumerator

C

2

1+C

3

−

1−C

3

Given :

\bf{\dfrac{\sqrt{1+C^3}-\sqrt{1-C^3}}{C^2}}

C

2

1+C

3

−

1−C

3

Solution :

\bf{Multiply\:by\:the\:conjugate}\:\dfrac{\sqrt{1+C^3}+\sqrt{1-C^3}}{\sqrt{1+C^3}+\sqrt{1-C^3}}Multiplybytheconjugate

1+C

3

+

1−C

3

1+C

3

+

1−C

3

✯ To rationlize the numerator, multiply the numerator and denominator

by conjugate of the radical : \bf{\sqrt{1+C^{3}}-\sqrt{1-C^{3}}

=\tt{\dfrac{\left(\sqrt{1+C^3}-\sqrt{1-C^3}\right)\left(\sqrt{1+C^3}+\sqrt{1-C^3}\right)}{C^2\left(\sqrt{1+C^3}+\sqrt{1-C^3}\right)}}=

C

2

(

1+C

3

+

1−C

3

)

(

1+C

3

−

1−C

3

)(

1+C

3

+

1−C

3

)

➜ Simplify the numerator

\bf{\left(\sqrt{1+C^3}-\sqrt{1-C^3}\right)\left(\sqrt{1+C^3}+\sqrt{1-C^3}\right)}(

1+C

3

−

1−C

3

)(

1+C

3

+

1−C

3