Question P.6.12: Simplifying a Fractional Expression Containing Radicals Simp...
Simplifying a Fractional Expression Containing Radicals
Simplify: \frac{\sqrt{9 – x²}+ \frac{x²}{\sqrt{9 – x²}}}{9 – x²}.
\frac{\sqrt{9 – x²}+ \frac{x²}{\sqrt{9 – x²}}}{9 – x²}
The least common denominator is \sqrt{9 – x²}.
=\frac{\sqrt{9 – x²}+ \frac{x²}{\sqrt{9 – x²}}}{9 – x²} · \frac{\sqrt{9 – x²}}{\sqrt{9 – x²}}
Multiply the numerator and the denominator by \sqrt{9 – x²}.
=\frac{\sqrt{9 – x²}\sqrt{9 – x²}+ \frac{x²}{\sqrt{9 – x²}}\sqrt{9 – x²}}{(9 – x²)\sqrt{9 – x²}}
Use the distributive property in the numerator.
=\frac{(9 – x²) + x²}{(9 – x²)^{\frac{3}{2}}}
In the denominator: (9 – x²)^1(9 – x²)^{\frac{1}{2}}=(9 – x²)^{1+\frac{1}{2}}=(9 – x²)^{\frac{3}{2}}.
=\frac{9}{\sqrt{(9-x^2)^3}}
Because the original expression was in radical form, write the denominator in radical form.