To rationalize the denominator of 4/(2+√8), Brittany multiplied by (2-√8)/(2-√8) and Justin multiplied by (1-√2)/(1-√2). Explain why both are correct.
To rationalize the denominator of a fraction, we want to eliminate any irrational numbers (like square roots) from the denominator. In this case, the denominator is 2+√8.
To explain why both Brittany's and Justin's approaches are correct, let's go through each of their methods.
Brittany decided to multiply the fraction by (2-√8)/(2-√8). To understand why this works, let's check the denominator after multiplying:
(2+√8) * (2-√8) = 4 - 2√8 + 2√8 - (√8 * -√8) = 4 - (√8 * -√8) = 4 - 8 = -4
Since the denominator is now -4, which is not an irrational number, Brittany's approach is valid and she has successfully rationalized the denominator.
Now let's analyze Justin's method. He decided to multiply the fraction by (1-√2)/(1-√2). After multiplying, the denominator becomes:
(2+√8) * (1-√2) = 2 - 2√2 + √8 - (√2 * √8) = 2 - 2√2 + √8 - √(2 * 8) = 2 - 2√2 + √8 - √16 = 2 - 2√2 + √8 - 4 = -2√2 - 2
In this case, the denominator is -2√2 - 2, which is another valid rationalized form of the denominator. So Justin's approach is also correct.
Both Brittany and Justin's methods are acceptable ways to rationalize the denominator of 4/(2+√8). They chose different expressions to multiply by, but both expressions effectively eliminated the irrational number from the denominator.