Question

rationalize the denominator and simplify.
\\(\frac{-5}{3\sqrt{3} + 4}\\)

Explanation:

Step1: Identify conjugate of denominator

The denominator is \(3\sqrt{3} + 4\), its conjugate is \(3\sqrt{3}-4\). Multiply numerator and denominator by the conjugate.
\[
\frac{-5}{3\sqrt{3} + 4}\times\frac{3\sqrt{3}-4}{3\sqrt{3}-4}
\]

Step2: Multiply numerators and denominators

Numerator: \(-5\times(3\sqrt{3}-4)=-15\sqrt{3}+20\)
Denominator: \((3\sqrt{3} + 4)(3\sqrt{3}-4)=(3\sqrt{3})^2-4^2 = 27 - 16=11\)

Step3: Write the simplified form

\[
\frac{-15\sqrt{3}+20}{11}=\frac{20 - 15\sqrt{3}}{11}
\]

Answer:

\(\frac{20 - 15\sqrt{3}}{11}\)