Operations with radical expressions: Simplify each quotient.
-3/sq rt 2 + sq rt 5
To simplify a quotient of radical expressions, we need to rationalize the denominator.
In this case, the denominator is: √2 + √5.
To rationalize it, we multiply both the numerator and the denominator by the conjugate of the denominator, which is formed by changing the sign between the two terms. So, the conjugate of √2 + √5 is √2 - √5.
Multiplying the numerator and denominator by √2 - √5, we get:
(-3/√2 + √5) * (√2 - √5) / (√2 + √5) * (√2 - √5)
Using the distributive property, we multiply the two terms in the numerator and the two terms in the denominator:
= (-3√2 + 3√5 + 3√2 - 3√(5^2)) / (√(2^2) - √(5^2))
Simplifying further, we have:
= (-3√2 + 3√5 + 3√2 - 3(5)) / (2 - 5)
= (-6 + 3√2 +3√5) / -3
Dividing each term by -3, we get:
= -6/-3 + (3√2)/-3 + (3√5)/-3
= 2 - (√2)/3 - (√5)/3
Thus, the simplified quotient is: 2 - (√2)/3 - (√5)/3.