($\sqrt{a} +\sqrt{8a})^2$ = 54 + b$\sqrt{2}$, a and b are positive integers. Find the value of a and the value of b.

1. Analyze the question

We are given the equation $(\sqrt{a} + \sqrt{8a})^2 = 54 + b\sqrt{2}$. We need to expand the left side, simplify it, and compare it to the right side to find the values of the positive integers $a$ and $b$.

2. Explain the concept

To solve this, we use the algebraic identity $(x + y)^2 = x^2 + 2xy + y^2$. We also need to remember the properties of surds:

• $\sqrt{x} \cdot \sqrt{x} = x$
• $\sqrt{8a} = \sqrt{4 \cdot 2 \cdot a} = 2\sqrt{2a}$

3. Show the solution

Step 1: Simplify the expression inside the bracket. $\sqrt{8a} = \sqrt{4 \times 2 \times a} = 2\sqrt{2a}$. So, the expression becomes: $(\sqrt{a} + 2\sqrt{2a})^2$

Step 2: Expand using $(x + y)^2 = x^2 + 2xy + y^2$. Let $x = \sqrt{a}$ and $y = 2\sqrt{2a}$.

• $x^2 = (\sqrt{a})^2 = a$
• $y^2 = (2\sqrt{2a})^2 = 4 \times 2a = 8a$
• $2xy = 2(\sqrt{a})(2\sqrt{2a}) = 4\sqrt{2a^2} = 4a\sqrt{2}$

Step 3: Combine the terms. $a + 8a + 4a\sqrt{2} = 9a + 4a\sqrt{2}$

Step 4: Compare with the given equation. $9a + 4a\sqrt{2} = 54 + b\sqrt{2}$

By comparing the rational parts and the irrational parts:

• Rational part: $9a = 54 \implies a = \frac{54}{9} \implies \mathbf{a = 6}$
• Irrational part: $4a\sqrt{2} = b\sqrt{2} \implies b = 4a$

Substitute $a = 6$ into the equation for $b$:

• $b = 4(6) \implies \mathbf{b = 24}$

4. Identify the answer

Comparing our results ($a=6, b=24$) with the given options, we find that it matches option C.

5. Tip

When dealing with surds in equations, always simplify the square roots (e.g., $\sqrt{8a} \to 2\sqrt{2a}$) before expanding. This makes it much easier to group the "rational" and "irrational" parts on both sides of the equation.