Rationalize the denominator and simplify. All variables represent positive real

Nann 2021-09-20 Answered
Rationalize the denominator and simplify. All variables represent positive real numbers.
\(\displaystyle{\frac{{{3}\sqrt{{{y}}}}}{{{2}\sqrt{{{x}}}-{3}\sqrt{{{y}}}}}}\)

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Expert Answer

Isma Jimenez
Answered 2021-09-21 Author has 18039 answers

Rationalization is used to eliminate root radical in the denominator, for this both the numerator and denominator is multiplied by algebraic conjugate of denominator .For example if the denominator of an expression is \(\displaystyle{a}+\sqrt{{{b}}}\) then the algebraic conjugate is \(\displaystyle{a}-\sqrt{{{b}}}\). Now conjugate is multiplied to both numerator and denominator to get rationalized form of the expression.
In the given expression \(\displaystyle{\frac{{{3}\sqrt{{{y}}}}}{{{2}\sqrt{{{x}}}-{3}\sqrt{{{y}}}}}}\) the conjugate of denominator \(\displaystyle{\left({2}\sqrt{{{x}}}-{3}\sqrt{{{y}}}\right)}\) is \((2\sqrt{x}+3\sqrt{y})\)
To get the rationalize form multiply the numerator and denominator by \(\displaystyle{\left({2}\sqrt{{{x}}}+{3}\sqrt{{{y}}}\right)}\)
Use the algebraic identity \(\displaystyle{\left({a}+{b}\right)}{\left({a}-{b}\right)}={a}^{{2}}-{b}^{{2}}\) to simplify the denominator
Apply product rule of exponent \(\displaystyle{a}^{{m}}\cdot{a}^{{n}}={a}^{{{m}+{n}}}\)
\(\displaystyle{\frac{{{3}\sqrt{{{y}}}}}{{{2}\sqrt{{{x}}}-{3}\sqrt{{{y}}}}}}={\frac{{{3}\sqrt{{{y}}}{\left({2}\sqrt{{{x}}}+{3}\sqrt{{{y}}}\right)}}}{{{\left({2}\sqrt{{{x}}}-{3}\sqrt{{{y}}}\right)}{\left({2}\sqrt{{{x}}}+{3}\sqrt{{{y}}}\right)}}}}\)
\(\displaystyle={\frac{{{6}\sqrt{{{x}{y}}}+{9}{y}}}{{{4}{x}-{9}{y}}}}\)

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