What is the following quotient 2 – sqrt 8? This question might seem straightforward at first glance, but it involves some interesting mathematical concepts. In this article, we will explore the properties of square roots and how to simplify the given quotient to find its value.
The expression 2 – sqrt 8 can be simplified by breaking down the square root of 8 into its prime factors. To do this, we need to find the largest perfect square that divides 8. In this case, it is 4, as 4 x 2 = 8. Therefore, we can rewrite sqrt 8 as sqrt(4 x 2), which is equal to sqrt 4 x sqrt 2.
Since sqrt 4 is equal to 2, the expression 2 – sqrt 8 becomes 2 – 2 x sqrt 2. Now, we can factor out a 2 from both terms in the expression:
2 – 2 x sqrt 2 = 2(1 – sqrt 2)
At this point, we have simplified the expression, but we can further simplify it by rationalizing the denominator. To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is 1 + sqrt 2. This gives us:
2(1 – sqrt 2) / (1 + sqrt 2) x (1 – sqrt 2) / (1 – sqrt 2)
Expanding the numerator and denominator, we get:
2(1 – 2sqrt 2 + 2) / (1 – 2 + 2)
Simplifying further, we have:
2(3 – 2sqrt 2) / 1
This means that the quotient 2 – sqrt 8 simplifies to 2(3 – 2sqrt 2). To find the numerical value of this expression, we can expand it:
2 x 3 – 2 x 2sqrt 2 = 6 – 4sqrt 2
Therefore, the value of the quotient 2 – sqrt 8 is 6 – 4sqrt 2. This shows that, while the original expression may have seemed complicated, it can be simplified using the properties of square roots and rationalizing denominators.
