How do you simplify (sqrta- sqrtb)/(sqrta+sqrtb)√a−√b√a+√b?

3 Answers
smendyka
Mar 14, 2018

See a solution process below:

Explanation:

To simplify we need to rationalize the denominator by multiplying by the appropriate form of 11:

(color(red)(sqrt(a)) - color(red)(sqrt(b)))/(color(red)(sqrt(a)) - color(red)(sqrt(b))) xx (sqrt(a) - sqrt(b))/(sqrt(a) + sqrt(b)) =>√a−√b√a−√b×√a−√b√a+√b⇒

(sqrt(a) - sqrt(b))^2/(color(red)(sqrt(a))sqrt(a) + color(red)(sqrt(a))sqrt(b) - color(red)(sqrt(b))sqrt(a) - color(red)(sqrt(b))sqrt(b)) =>(√a−√b)2√a√a+√a√b−√b√a−√b√b⇒

(sqrt(a) - sqrt(b))^2/(a - b)(√a−√b)2a−b

Jade
Mar 14, 2018

(sqrta-sqrtb)/(sqrta+sqrtb)√a−√b√a+√b

Multiply both the numerator and denominator by sqrta-sqrtb√a−√b

(sqrta-sqrtb)/(sqrta+sqrtb)*(sqrta-sqrtb)/(sqrta-sqrtb)√a−√b√a+√b⋅√a−√b√a−√b

(sqrta-sqrtb)^2/((sqrta)^2-(sqrtb)^2)(√a−√b)2(√a)2−(√b)2

(a+b-2sqrt(ab))/(a-b)a+b−2√aba−b

Hope this helps :)

Jim G.
Mar 14, 2018

(a-2sqrt(ab)+b)/(a-b)a−2√ab+ba−b

Explanation:

"multiply the numerator/denominator by the "color(blue)"conjugate"multiply the numerator/denominator by the conjugate
"of the denominator"of the denominator

"the conjugate of "sqrta+sqrtb" is "sqrtacolor(red)(-)sqrtbthe conjugate of √a+√b is √a−√b

"this ensures the denominator is a rational value"this ensures the denominator is a rational value

•color(white)(x)sqrtaxxsqrta=a∙x√a×√a=a

•color(white)(x)(sqrta+sqrtb)(sqrta-sqrtb)=a-b∙x(√a+√b)(√a−√b)=a−b

rArr(sqrta-sqrtb)/(sqrta+sqrtb)xx(sqrta-sqrtb)/(sqrta-sqrtb)⇒√a−√b√a+√b×√a−√b√a−√b

=((sqrta-sqrtb)(sqrta-sqrtb))/((sqrta+sqrtb)(sqrta-sqrtb))=(√a−√b)(√a−√b)(√a+√b)(√a−√b)

=(a-sqrtab-sqrtab+b)/(a-b)=a−√ab−√ab+ba−b

=(a-2sqrt(ab)+b)/(a-b)=a−2√ab+ba−b

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