Simplifying Radical Expressions

The goal is to simplify the given expression $\frac{3\sqrt{5} - 5}{3\sqrt{5} + 5}$ and express it in the form $a + b\sqrt{5}$ to find the value of $b$.

Rationalizing the Denominator

To simplify the fraction, we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is $3\sqrt{5} - 5$.

Expanding and Simplifying

Now, expand the numerator and the denominator:

• Numerator: $(3\sqrt{5} - 5)^2$
  • Using the formula $(x - y)^2 = x^2 - 2xy + y^2$:
  $ (3\sqrt{5})^2 - 2(3\sqrt{5})(5) + 5^2 $ $ (9 \times 5) - 30\sqrt{5} + 25 $ $ 45 - 30\sqrt{5} + 25 = 70 - 30\sqrt{5} $
• Denominator: $(3\sqrt{5} + 5)(3\sqrt{5} - 5)$
  • Using the formula $(x + y)(x - y) = x^2 - y^2$:
  $ (3\sqrt{5})^2 - 5^2 $ $ (9 \times 5) - 25 $ $ 45 - 25 = 20 $

Final Expression

The simplified fraction is:

Separate the terms to match the form $a + b\sqrt{5}$:

Simplify the fractions:

Identifying the Value of b

Compare the simplified expression $\frac{7}{2} - \frac{3}{2}\sqrt{5}$ with the given form $a + b\sqrt{5}$.

• $a = \frac{7}{2}$
• $b = -\frac{3}{2}$

Therefore, the value of $b$ is $-\frac{3}{2}$.