Multiplying by the conjugate—(√7 + √3) on top and bottom—is allowed because it's like multiplying by one, which doesn't change the value. You do it to use the difference of squares, canceling the roots in the denominator. So, 1/(√7 - √3) becomes (√7 + √3)/4.
My engineering prep makes me see it clearly: rules are tools, just like proper form in the gym. You don't ask *why* a spotter helps—you just know it prevents injury and lets you lift heavier. Same here: the conjugate is your spotter to simplify the problem.
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