Simplify: √3 / (1 - √5)
Understand the Problem
The question asks to simplify an expression involving a fraction with a square root in the numerator and a difference involving a square root in the denominator. Simplification typically involves rationalizing the denominator, which means eliminating the square root from the denominator. This is achieved by multiplying both the numerator and denominator by the conjugate of the denominator.
Answer
$\frac{\sqrt{15}+\sqrt{6}}{3}$
Answer for screen readers
$\frac{\sqrt{15}+\sqrt{6}}{3}$
Steps to Solve
- Identify the conjugate of the denominator
The denominator is $\sqrt{5} - \sqrt{2}$. Its conjugate is $\sqrt{5} + \sqrt{2}$.
- Multiply the numerator and denominator by the conjugate
Multiply the given expression by $\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}+\sqrt{2}}$:
$$ \frac{\sqrt{3}}{\sqrt{5}-\sqrt{2}} \cdot \frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}+\sqrt{2}} $$
- Simplify the numerator
Multiply $\sqrt{3}$ by $(\sqrt{5}+\sqrt{2})$:
$$ \sqrt{3}(\sqrt{5}+\sqrt{2}) = \sqrt{3}\cdot\sqrt{5} + \sqrt{3}\cdot\sqrt{2} = \sqrt{15} + \sqrt{6} $$
- Simplify the denominator
Multiply $(\sqrt{5}-\sqrt{2})$ by $(\sqrt{5}+\sqrt{2})$:
$$ (\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2}) = (\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3 $$
- Write the simplified expression
Combine the simplified numerator and denominator:
$$ \frac{\sqrt{15}+\sqrt{6}}{3} $$
$\frac{\sqrt{15}+\sqrt{6}}{3}$
More Information
Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction, which is often useful in simplifying expressions and performing further calculations.
Tips
A common mistake is to incorrectly compute the product of the conjugates in the denominator. Remember that $(a-b)(a+b) = a^2 - b^2$. Also, be careful distributing and multiplying the terms in the numerator.
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