Understanding the Problem: Rationalizing Surds and Finding a + b

The problem asks us to find the value of \(a + b\) given an equation where a fraction with surds in the denominator is equal to an expression of the form \(a\sqrt 3 - b\). To solve this, we first need to simplify the given fraction by rationalizing the denominator. Rationalizing the denominator helps us to rewrite the expression in a simpler form that can be easily compared to \(a\sqrt 3 - b\).

Step-by-Step Solution: Rationalizing the Denominator

The given equation is:

\(\frac {8 + 2\sqrt 3}{3\sqrt 3 + 5} = a\sqrt 3 - b\)

We will start by rationalizing the left side of the equation. The denominator is \(3\sqrt 3 + 5\). The conjugate of \(3\sqrt 3 + 5\) is \(3\sqrt 3 - 5\). We multiply both the numerator and the denominator by this conjugate.

LHS \( = \frac {8 + 2\sqrt 3}{3\sqrt 3 + 5} \times \frac {3\sqrt 3 - 5}{3\sqrt 3 - 5}\)

Multiplying the Numerator

Numerator \( = (8 + 2\sqrt 3)(3\sqrt 3 - 5)\)

We use the distributive property (or FOIL method) to multiply the terms:

• \(8 \times (3\sqrt 3) = 24\sqrt 3\)
• \(8 \times (-5) = -40\)
• \((2\sqrt 3) \times (3\sqrt 3) = 2 \times 3 \times \sqrt 3 \times \sqrt 3 = 6 \times 3 = 18\)
• \((2\sqrt 3) \times (-5) = -10\sqrt 3\)

Adding these terms:

Numerator \( = 24\sqrt 3 - 40 + 18 - 10\sqrt 3\)

Combine like terms (terms with \(\sqrt 3\) and constant terms):

Numerator \( = (24\sqrt 3 - 10\sqrt 3) + (-40 + 18)\)

Numerator \( = 14\sqrt 3 - 22\)

Multiplying the Denominator

Denominator \( = (3\sqrt 3 + 5)(3\sqrt 3 - 5)\)

This is in the form \((x+y)(x-y) = x^2 - y^2\), where \(x = 3\sqrt 3\) and \(y = 5\).

Denominator \( = (3\sqrt 3)^2 - 5^2\)

Calculate \((3\sqrt 3)^2\):

\((3\sqrt 3)^2 = 3^2 \times (\sqrt 3)^2 = 9 \times 3 = 27\)

Calculate \(5^2\):

\(5^2 = 25\)

Denominator \( = 27 - 25 = 2\)

Simplifying the Fraction

Now we have the simplified numerator and denominator:

LHS \( = \frac{14\sqrt 3 - 22}{2}\)

We can divide each term in the numerator by the denominator:

LHS \( = \frac{14\sqrt 3}{2} - \frac{22}{2}\)

LHS \( = 7\sqrt 3 - 11\)

Comparing and Finding a and b

Now we compare the simplified left side with the right side of the original equation:

\(7\sqrt 3 - 11 = a\sqrt 3 - b\)

By comparing the coefficients of \(\sqrt 3\) and the constant terms on both sides, we can find the values of \(a\) and \(b\).

• Coefficient of \(\sqrt 3\): \(a = 7\)
• Constant term: \(-b = -11\), which means \(b = 11\)

Calculating a + b

The problem asks for the value of \(a + b\).

\(a + b = 7 + 11\)

\(a + b = 18\)

Thus, the value of \(a + b\) is 18.

Revision Table: Key Concepts

Additional Information: Surds and Rational Numbers

Surds are irrational numbers that can be expressed as the root of a number, like \(\sqrt 2\), \(\sqrt 3\), \(\sqrt[3] 5\), etc. A number is a surd if its root cannot be simplified into a rational number. For example, \(\sqrt 4\) is not a surd because \(\sqrt 4 = 2\), which is a rational number.

A rational number is any number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\). Examples include \(1\), \(-5\), \(1/2\), \(0.75\).

When we rationalize a denominator containing surds, we are essentially changing the form of the expression so that the denominator becomes a rational number, making it easier to work with or compare.

In this problem, after rationalizing, the expression \(7\sqrt 3 - 11\) is in the form \(a\sqrt 3 - b\), where \(a=7\) and \(b=11\). Both \(a\) and \(b\) are rational numbers (integers, specifically), which is typically expected in such problems unless otherwise stated.