Simplify the following expression: $\frac{4\sqrt{x}}{\sqrt{x+1} - \sqrt{x}}$
Understand the Problem
The question requires simplifying the given expression, which involves a fraction with radicals in both the numerator and denominator. The key is likely to rationalize the denominator to simplify the expression.
Answer
$4x + 4\sqrt{x(x+1)}$
Answer for screen readers
$4x + 4\sqrt{x(x+1)}$
Steps to Solve
- Identify the conjugate of the denominator
The conjugate of the denominator $\sqrt{x+1} - \sqrt{x}$ is $\sqrt{x+1} + \sqrt{x}$.
- Multiply the numerator and denominator by the conjugate
Multiply both the numerator and the denominator of the given expression by the conjugate:
$$ \frac{4\sqrt{x}}{\sqrt{x+1} - \sqrt{x}} \cdot \frac{\sqrt{x+1} + \sqrt{x}}{\sqrt{x+1} + \sqrt{x}} $$
- Simplify the denominator
Using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$, simplify the denominator:
$$ (\sqrt{x+1} - \sqrt{x})(\sqrt{x+1} + \sqrt{x}) = (\sqrt{x+1})^2 - (\sqrt{x})^2 = (x+1) - x = 1 $$
- Simplify the numerator
Distribute $4\sqrt{x}$ in the numerator:
$$ 4\sqrt{x}(\sqrt{x+1} + \sqrt{x}) = 4\sqrt{x}\sqrt{x+1} + 4\sqrt{x}\sqrt{x} = 4\sqrt{x(x+1)} + 4x $$
- Write the simplified expression
Since the denominator is 1, the simplified expression is just the simplified numerator:
$$ \frac{4\sqrt{x(x+1)} + 4x}{1} = 4\sqrt{x(x+1)} + 4x $$
$4x + 4\sqrt{x(x+1)}$
More Information
The expression has been simplified by rationalizing the denominator, which eliminates the radicals from the denominator.
Tips
A common mistake is not correctly identifying or applying the conjugate. Also, errors in simplification, especially when dealing with radicals and distribution, can occur. It is important to carefully perform each step to avoid these mistakes.
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