Solution
We want to rewrite the expression
$$ \frac{a - \sqrt{48}}{\sqrt{3} + 1} $$
in the form $b\sqrt{3} - 9$, where $a$ and $b$ are integers.
Step 1: Simplify the square root and the fraction
First, simplify $\sqrt{48}$:
$$ \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} $$
Thus, the expression becomes:
$$ \frac{a - 4\sqrt{3}}{\sqrt{3} + 1} $$
Step 2: Rationalize the denominator
To rationalize the denominator, multiply numerator and denominator by $\sqrt{3} - 1$:
$$ \frac{a - 4\sqrt{3}}{\sqrt{3} + 1} \cdot \frac{\sqrt{3} - 1}{\sqrt{3} - 1} $$
Simplify the denominator:
$$ (\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2 $$
The numerator becomes:
$$ (a - 4\sqrt{3})(\sqrt{3} - 1) $$
Expand this:
$$ (a - 4\sqrt{3})(\sqrt{3} - 1) = a\sqrt{3} - a - 4\sqrt{3}\cdot\sqrt{3} + 4\sqrt{3} $$
Simplify each term:
$$ a\sqrt{3} - a - 4(3) + 4\sqrt{3} = a\sqrt{3} - a - 12 + 4\sqrt{3} $$
Combine like terms:
$$ (a + 4)\sqrt{3} - (a + 12) $$
Thus, the numerator becomes:
$$ (a + 4)\sqrt{3} - (a + 12) $$
Step 3: Simplify the fraction
Divide the numerator by 2 (the denominator):
$$ \frac{(a + 4)\sqrt{3} - (a + 12)}{2} = \frac{a + 4}{2}\sqrt{3} - \frac{a + 12}{2} $$
Step 4: Match the form $b\sqrt{3} - 9$
We are told the expression can be written as:
$$ b\sqrt{3} - 9 $$
Equating coefficients, we get:
$$ \frac{a + 4}{2} = b \quad \text{and} \quad \frac{a + 12}{2} = 9 $$
Step 5: Solve for $a$ and $b$
From
$$ \frac{a + 12}{2} = 9 $$
we get:
$$ a + 12 = 18 \implies a = 6 $$
Substitute $a = 6$ into
$$ \frac{a + 4}{2} = b $$
$$ \frac{6 + 4}{2} = b \implies \frac{10}{2} = b \implies b = 5 $$
Final Answer
$$ a = 6, \quad b = 5 $$
Post a Comment