math_new / train /algebra /1045.json
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{
"problem": "Rationalize the denominator: $\\frac{1}{\\sqrt{2}-1}$. Express your answer in simplest form.",
"level": "Level 3",
"type": "Algebra",
"solution": "To get the square root out of the denominator, we can multiply the numerator and denominator by $(\\sqrt{2}+1)$ so that the $\\sqrt{2}$ is squared and $\\sqrt{2}$ and $-\\sqrt{2}$ cancel each other out. $$\\frac{1}{\\sqrt{2}-1}\\cdot\\frac{\\sqrt{2}+1}{\\sqrt{2}+1}=\\frac{\\sqrt{2}+1}{2-\\sqrt{2}+\\sqrt{2}-1}=\\frac{\\sqrt{2}+1}{1}=\\boxed{\\sqrt{2}+1}$$"
}