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I have a question in my text book that asks me to simplify the above expression.

I reached the solution:

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However, the textbook gives the answer as:

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I appreciate that the value of these two expressions is the same. However, I am unsure as to how and why the answer above was reached instead of my answer.

Troy
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0.5772156649
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  • It is customary to rationalize the denominator in situations like these, so you just need to multiply the numerator and denominator by the conjugate $\sqrt{a}+\sqrt{b}$. – Troy Dec 29 '16 at 13:36
  • It might be worth editing the Question to indicate that $a,b$ are assumed to be distinct positive real numbers (or distinct nonnegative real numbers, and not both zero). – hardmath Dec 29 '16 at 13:45

6 Answers6

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Multiply your solution by $\dfrac{\sqrt a + \sqrt b}{\sqrt a + \sqrt b}$ and simplify.

lhf
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Multiply your solution by $\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}}$, you can get the textbook answer.

It is because we usually like the denominator to be rational.

Gary Tam
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multiplying denominator and numerator by $$\sqrt{a}+\sqrt{b}$$ we get $$\frac{a\sqrt{ab}(\sqrt{a}+\sqrt{b})}{a-b}$$

Dr. Sonnhard Graubner
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Hint:

$$\frac{a\sqrt{ab}}{\sqrt{a}-\sqrt{b}} = \frac{a\sqrt{ab}}{\sqrt{a}-\sqrt{b}}\cdot\frac{\sqrt a + \sqrt b}{\sqrt a + \sqrt b}$$

Now, simplify the expression. Remember, $(x-y)(x+y)=x^2-y^2$

5xum
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Typically, the denominator does not carry a root. Therefore, we apply$(x-y)(x+y)=x^2-y^2$to the denominator. Then, numerator and denominator multiply $\sqrt{a}+\sqrt{b}$. We get the answer $$\frac{a^2\sqrt{b}+ab\sqrt{a}}{a-b}$$. P.S. The answer is a bit ugly.

郑豆浆
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What your textbook did is called rationalization. In which if we want to remove radicals($\sqrt.$) then whole expression is multiplied and divided by a conjugate of either numerator or denominator.

If you don't want radicals in numerator then divide whole expression with conjugate of numerator and if you don't want radicals in denominator then multiply and divide whole with conjugate of denominator.

Here you can learn more about it.

$$\frac{a\sqrt{ab}}{\sqrt{a}-\sqrt{b}} = \frac{a\sqrt{ab}}{\sqrt{a}-\sqrt{b}}\cdot\frac{\sqrt a + \sqrt b}{\sqrt a + \sqrt b}$$

Fawad
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