Question

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

Explanation:

Step1: Identify the conjugate

The denominator is \(4\sqrt{3} + 4\), so its conjugate is \(4\sqrt{3}-4\). We multiply the numerator and denominator by this conjugate.

$$ \frac{5}{4\sqrt{3}+4}\times\frac{4\sqrt{3}-4}{4\sqrt{3}-4} $$

Step2: Multiply numerator and denominator

Multiply the numerators: \(5\times(4\sqrt{3}-4)=20\sqrt{3}-20\)
Multiply the denominators using the difference of squares formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a = 4\sqrt{3}\) and \(b = 4\).

$$ (4\sqrt{3})^{2}-4^{2}=16\times3 - 16=48 - 16 = 32 $$

Step3: Simplify the fraction

Now we have \(\frac{20\sqrt{3}-20}{32}\). We can factor out a 4 from the numerator: \(\frac{4(5\sqrt{3}-5)}{32}\), and then simplify by dividing numerator and denominator by 4.

$$ \frac{5\sqrt{3}-5}{8} $$

Answer:

\(\frac{5\sqrt{3}-5}{8}\)